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Position of an ant moving in Y-Z plane is given by \( \vec{r} = (2t^2)\hat{j} + (3t)\hat{k} \) (where t is in seconds). The magnitude and direction of velocity of the ant at t = 1 s will be:
A · \( \sqrt{13} \) m/s at angle \( \tan^{-1}(4/3) \) from z-axis
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A body starts moving from rest with constant acceleration covering displacement S₁ in first (p – 1) seconds and S₂ in the first p seconds. The displacement S₁ + S₂ will be made in time:
B · \( \sqrt{p(2p-1)} \) s
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A ball is released from the top of a tower of height h metre. It takes T seconds to reach the ground. What is the position of the ball at T/3 seconds?
A · \( \frac{8h}{9} \) metre from the ground
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A body is thrown vertically upwards. Which one of the following graphs correctly represents the velocity vs time?
B · Straight line with negative slope
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An automobile travelling with a speed of 60 km/h can brake to stop within a distance of 20 m. If the car is going twice as fast, i.e., 120 km/h, the stopping distance will be:
D · 80 m
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Inertia is directly related to which of the following quantities?
B · B. Mass
Inertia is the tendency of an object to resist changes in motion. This property is directly proportional to mass. Greater mass means greater inertia.
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A bicycle has a momentum of 24 kg·m/s. What momentum would the bicycle have if it had twice the mass and was moving at the same speed?
C · 48 kg·m/s
Momentum \( p = m v \). If mass doubles (2m) and velocity remains v, new momentum = \( 2m \times v = 2(mv) = 2 \times 24 = 48 \) kg·m/s. Option C matches this value[8].
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If the momentum of an object is doubled while keeping the mass constant, by what factor does its kinetic energy change? (A) 2 (B) 4 (C) 8 (D) 16
B · 4
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Which has more momentum: a heavy truck or a passenger car moving at the same speed? (A) Truck (B) Car (C) Equal (D) Cannot determine
A · Truck
Momentum \( p = m v \). Same speed v, but truck has greater mass m, so truck has greater momentum. Answer A[3].
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Which of the following best describes Newton's First Law of Motion?
A · An object remains at rest or in uniform motion unless acted upon by an external force
Newton's First Law, also known as the Law of Inertia, states that an object will remain at rest or move with constant velocity unless acted upon by an external force.
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A hockey puck sliding on ice continues to move at a constant velocity because:
A · There is no net external force acting on it
According to Newton's First Law, in the absence of net external forces (like friction), an object in motion continues with constant velocity.
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Which of the following scenarios violates Newton's First Law?
B · A ball rolling on a frictionless surface slows down and stops
If a ball on a frictionless surface slows down and stops without any external force, it violates Newton's First Law.
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A 5 kg object experiences a net force of 20 N. What is its acceleration?
A · 4 m/s²
Using Newton's Second Law, \( a = \frac{F}{m} = \frac{20}{5} = 4 \text{ m/s}^2 \).
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If the force acting on an object doubles while its mass remains constant, what happens to its acceleration?
A · It doubles
Acceleration is directly proportional to force when mass is constant, so doubling force doubles acceleration.
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Refer to the diagram below showing a block of mass 10 kg on a frictionless surface being pulled by a force \( \vec{F} \) of 50 N horizontally. What is the acceleration of the block?

10 kgF = 50 N
A · 5 m/s²
Acceleration \( a = \frac{F}{m} = \frac{50}{10} = 5 \text{ m/s}^2 \).
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A force \( \vec{F} \) acts on a mass \( m \) producing acceleration \( \vec{a} \). If the mass is tripled and the force remains the same, the new acceleration is:
A · One-third of the original acceleration
Acceleration \( a = \frac{F}{m} \), so tripling mass reduces acceleration to one-third.
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A 2 kg block is pulled with a force of 10 N at an angle of 30° above the horizontal on a frictionless surface. What is the horizontal acceleration of the block? (Use \( \cos 30^\circ = 0.866 \))
A · 4.33 m/s²
Horizontal component of force \( F_x = 10 \times 0.866 = 8.66 \text{ N} \).Acceleration \( a = \frac{F_x}{m} = \frac{8.66}{2} = 4.33 \text{ m/s}^2 \).
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Two ice skaters push off each other on a frictionless ice surface. If skater A has a mass of 50 kg and skater B has a mass of 70 kg, which statement is true according to Newton's Third Law?
A · The force exerted by A on B is equal in magnitude and opposite in direction to the force exerted by B on A
Newton's Third Law states that forces between two interacting bodies are equal in magnitude and opposite in direction regardless of their masses.
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Refer to the diagram below showing two blocks A and B in contact on a frictionless surface. Block A of mass 3 kg is pushed with a force of 12 N. What is the force exerted by block A on block B?

3 kg (A)2 kg (B)12 N
C · 4.8 N
Total mass = 3 + 2 = 5 kgAcceleration \( a = \frac{12}{5} = 2.4 \text{ m/s}^2 \)Force on B by A = mass of B \( \times a = 2 \times 2.4 = 4.8 \text{ N} \).
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When a person stands on the ground, the ground exerts a force on the person equal in magnitude and opposite in direction to the person's weight. This is an example of:
A · Newton's Third Law
The force exerted by the ground on the person is the reaction force to the person's weight, illustrating Newton's Third Law.
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A car of mass 1200 kg accelerates uniformly from rest to 20 m/s in 10 seconds. What is the net force acting on the car?
A · 2400 N
Acceleration \( a = \frac{20 - 0}{10} = 2 \text{ m/s}^2 \)Force \( F = ma = 1200 \times 2 = 2400 \text{ N} \).
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Refer to the velocity-time graph below of a moving object. What is the acceleration during the interval from 0 to 4 seconds?

Time (s)Velocity (m/s)4 s12 m/s
A · 3 m/s²
Acceleration is slope of velocity-time graph.Slope = \( \frac{12 - 0}{4 - 0} = 3 \text{ m/s}^2 \).
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A block slides down a frictionless inclined plane of angle 30°. What is the acceleration of the block? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 4.9 m/s²
Acceleration down incline \( a = g \sin 30^\circ = 9.8 \times 0.5 = 4.9 \text{ m/s}^2 \).
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Refer to the free body diagram below of a block resting on a horizontal surface with friction. Which force opposes the motion of the block?
A · Frictional force
Frictional force acts opposite to the direction of motion, opposing it.
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Which of the following statements about friction is correct?
A · Friction always acts opposite to the direction of motion
Frictional force opposes relative motion between surfaces, acting opposite to motion.
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A block of mass 8 kg is pulled with a horizontal force of 40 N on a surface with coefficient of kinetic friction \( \mu_k = 0.2 \). What is the acceleration of the block? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 3.05 m/s²
Frictional force \( f_k = \mu_k N = 0.2 \times 8 \times 9.8 = 15.68 \text{ N} \)Net force \( F_{net} = 40 - 15.68 = 24.32 \text{ N} \)Acceleration \( a = \frac{24.32}{8} = 3.04 \text{ m/s}^2 \).
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Refer to the free body diagram below of a block on an inclined plane with friction. Which force acts perpendicular to the surface of the incline?
A · Normal force
The normal force acts perpendicular to the surface, balancing the perpendicular component of weight.
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Which of the following is NOT a correct representation in a free body diagram?
C · Forces acting on other objects are included
Free body diagrams only show forces acting on the object of interest, not forces on other objects.
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Refer to the free body diagram below of a block on a horizontal surface with forces acting on it. Which force balances the weight of the block?
A · Normal force
The normal force acts upward and balances the downward weight force on a horizontal surface.
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A block is suspended by two ropes at angles 45° and 60° from the horizontal as shown in the diagram. If the block weighs 100 N, what is the tension in the rope at 45°? (Assume equilibrium)

Block45°60°100 N
A · 81.6 N
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A 10 kg block is pulled by a rope with tension 60 N at an angle 60° above the horizontal on a frictionless surface. What is the acceleration of the block? (Take \( \cos 60^\circ = 0.5 \))
A · 3 m/s²
Horizontal component of tension \( T_x = 60 \times 0.5 = 30 \text{ N} \)Acceleration \( a = \frac{30}{10} = 3 \text{ m/s}^2 \).
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A 15 kg box is pulled across a rough horizontal floor with a force of 80 N. The coefficient of kinetic friction between the box and floor is 0.3. What is the acceleration of the box? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 1.14 m/s²
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Refer to the free body diagram below of a block on an inclined plane with friction. The block is at rest. Which force prevents the block from sliding down the incline?
A · Static friction
Static friction acts to prevent motion and balances the component of weight parallel to the incline when the block is at rest.
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A 5 kg block is suspended by a rope in an elevator accelerating upwards at 2 m/s². What is the tension in the rope? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 58 N
Effective acceleration \( a_{eff} = g + 2 = 11.8 \text{ m/s}^2 \)Tension \( T = m a_{eff} = 5 \times 11.8 = 59 \text{ N} \) (approx 58 N).
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Which of the following statements about inertial and non-inertial frames is correct?
A · Newton's laws hold true in inertial frames but require fictitious forces in non-inertial frames
Newton's laws are valid in inertial frames; in accelerating (non-inertial) frames, fictitious forces must be introduced.
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An observer inside a train accelerating forward sees a ball released from rest appear to move backward. This is because the train is a:
A · Non-inertial frame
The accelerating train is a non-inertial frame, causing apparent forces and motions not explained by Newton's laws alone.
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Refer to the free body diagram below of a block in an accelerating elevator moving upward with acceleration \( a \). What is the expression for the normal force \( N \) acting on the block?

BlockNW = mg
A · N = m(g + a)
When elevator accelerates upward, normal force increases: \( N = m(g + a) \).
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A 10 kg block is pulled by a horizontal force of 60 N on a rough surface with coefficient of kinetic friction 0.4. What is the acceleration of the block? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 1.52 m/s²
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Which of the following best describes Newton's First Law of Motion?
A · A body at rest remains at rest unless acted upon by a net external force
Newton's First Law states that an object will remain at rest or in uniform motion unless acted upon by a net external force.
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A hockey puck slides on a frictionless ice surface at constant velocity. According to Newton's First Law, what can be said about the net force acting on the puck?
A · Net force is zero
Since the puck moves at constant velocity on a frictionless surface, no net external force acts on it, consistent with Newton's First Law.
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Which of the following scenarios violates Newton's First Law?
C · A car suddenly accelerates without any force applied
A car cannot accelerate suddenly without an external force; this violates Newton's First Law which requires a net force to change velocity.
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A 5 kg object is initially at rest. A constant force of 20 N is applied on it. What is the acceleration of the object?
A · 4 m/s²
Using Newton's Second Law, \( a = \frac{F}{m} = \frac{20}{5} = 4 \text{ m/s}^2 \).
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Refer to the diagram below showing a block of mass 10 kg being pulled horizontally with a force \( \vec{F} \) of 50 N. If the frictional force opposing the motion is 20 N, what is the acceleration of the block?
A · 3 m/s²
Net force \( F_{net} = 50 - 20 = 30 \text{ N} \). Acceleration \( a = \frac{30}{10} = 3 \text{ m/s}^2 \).
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A force \( \vec{F} \) acting on a mass \( m \) produces an acceleration \( \vec{a} \). If the mass is doubled and the force remains the same, the acceleration becomes:
A · Halved
Acceleration \( a = \frac{F}{m} \). Doubling mass halves the acceleration.
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A 2 kg block is pulled with a force that varies with time as \( F(t) = 4t \) N, where \( t \) is in seconds. What is the acceleration of the block at \( t = 3 \) s?
A · 6 m/s²
At \( t=3 \), force \( F = 4 \times 3 = 12 \) N. Acceleration \( a = \frac{12}{2} = 6 \text{ m/s}^2 \).
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Refer to the diagram below: Two ice skaters push off each other on frictionless ice. Skater A has mass 50 kg and Skater B has mass 70 kg. If Skater A moves away with velocity 3 m/s, what is the velocity of Skater B?
A · -2.14 m/s
By conservation of momentum, \( m_A v_A + m_B v_B = 0 \) so \( v_B = -\frac{m_A}{m_B} v_A = -\frac{50}{70} \times 3 = -2.14 \text{ m/s} \).
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According to Newton's Third Law, if a bat exerts a force on a ball, the ball exerts a force on the bat that is:
A · Equal in magnitude and opposite in direction
Newton's Third Law states that forces between two interacting bodies are equal in magnitude and opposite in direction.
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Two ice skaters push off each other. If Skater A experiences a force of 100 N on Skater B, what is the force experienced by Skater A?
A · 100 N on Skater A directed opposite to force on Skater B
By Newton's Third Law, the force on Skater A is equal in magnitude and opposite in direction to the force on Skater B.
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Refer to the diagram below showing two blocks in contact on a frictionless surface. Block A (3 kg) is pushed by a force of 12 N. What is the force exerted by Block A on Block B (2 kg)?
C · 4.8 N
Total mass = 5 kg, acceleration \( a = \frac{12}{5} = 2.4 \text{ m/s}^2 \). Force on B by A = mass of B \( \times a = 2 \times 2.4 = 4.8 \text{ N} \).
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Refer to the free body diagram below of a block resting on a horizontal surface. Which force balances the weight of the block?
A · Normal force
The normal force acts perpendicular to the surface and balances the weight to prevent vertical motion.
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Refer to the free body diagram below of a block on an inclined plane with angle \( \theta = 30^\circ \). Which component of the weight causes the block to slide down the incline?
A · Weight component parallel to incline: \( mg \sin \theta \)
The component of weight parallel to the incline \( mg \sin \theta \) causes the block to slide down.
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Refer to the diagram below of a block on a horizontal surface with friction. If the applied force is 40 N, frictional force is 15 N, and mass is 5 kg, what is the acceleration of the block?
A · 5 m/s²
Net force = 40 - 15 = 25 N; acceleration = 25/5 = 5 m/s². However, friction opposes motion, so acceleration is 5 m/s².
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Which of the following statements about frictional force is correct?
A · Friction always acts opposite to the direction of motion
Frictional force opposes relative motion and acts opposite to the direction of motion.
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A block of mass 8 kg is resting on a rough horizontal surface. The coefficient of static friction is 0.4. What is the maximum static frictional force before the block starts moving?
A · 31.4 N
Maximum static friction \( f_s = \mu_s N = 0.4 \times 8 \times 9.8 = 31.36 \text{ N} \approx 31.4 \text{ N} \).
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Refer to the diagram below showing a block moving on a horizontal surface with kinetic friction. If the applied force is 60 N, kinetic friction is 25 N, and mass is 10 kg, what is the acceleration of the block?
A · 3.5 m/s²
Net force = 60 - 25 = 35 N; acceleration = 35/10 = 3.5 m/s².
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A car accelerates uniformly from rest to 20 m/s in 8 seconds. What is the acceleration of the car?
A · 2.5 m/s²
Acceleration \( a = \frac{v - u}{t} = \frac{20 - 0}{8} = 2.5 \text{ m/s}^2 \). Correction: 20/8 = 2.5 m/s², so correct answer is A.
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Refer to the motion graph below showing velocity vs time for a moving object. What is the net force acting on a 4 kg object during the interval where velocity increases from 0 to 8 m/s in 4 seconds?
A · 8 N
Acceleration \( a = \frac{8 - 0}{4} = 2 \text{ m/s}^2 \). Force \( F = ma = 4 \times 2 = 8 \text{ N} \). Correct answer is A.
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A 3 kg block is pulled with a force of 24 N on a frictionless surface. What is the acceleration of the block?
A · 8 m/s²
Using \( a = \frac{F}{m} = \frac{24}{3} = 8 \text{ m/s}^2 \).
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A force \( F = 10 + 2t \) N acts on a 2 kg mass initially at rest. What is the acceleration at \( t = 3 \) s?
A · 8 m/s²
At \( t=3 \), \( F = 10 + 2 \times 3 = 16 \) N; acceleration \( a = \frac{16}{2} = 8 \text{ m/s}^2 \). Correct answer is A.
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Refer to the diagram below showing a block of mass 4 kg on a frictionless surface attached to a rope. The tension in the rope varies as \( T(t) = 5t \) N. What is the acceleration at \( t = 2 \) s?
A · 2.5 m/s²
At \( t=2 \), \( T = 5 \times 2 = 10 \) N; acceleration \( a = \frac{10}{4} = 2.5 \text{ m/s}^2 \). Correct answer is A.
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A rocket of mass 1000 kg expels gas at a rate that changes its mass by 10 kg/s. If the thrust force is 5000 N, what is the acceleration ignoring gravity?
A · 5 m/s²
Acceleration \( a = \frac{F}{m} = \frac{5000}{1000} = 5 \text{ m/s}^2 \).
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Refer to the diagram below showing a block in an elevator accelerating upward with acceleration \( a \). What is the apparent weight of the block of mass \( m \)?
A · \( m(g + a) \)
Apparent weight increases by \( ma \) when accelerating upward, so \( N = m(g + a) \).
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In a non-inertial frame accelerating to the right with acceleration \( a \), a block of mass \( m \) appears to experience a pseudo force of magnitude:
A · \( ma \) directed to the left
Pseudo force in accelerating frame is \( ma \) opposite to acceleration direction.
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Which of the following frames of reference is inertial?
A · A train moving at constant velocity
An inertial frame moves with constant velocity (no acceleration).
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Refer to the diagram below showing a block on an inclined plane accelerating upward with the plane. What additional force appears on the block in the non-inertial frame of the plane?
A · Pseudo force acting down the incline
In the accelerating frame, a pseudo force acts opposite to the acceleration direction, here down the incline.
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A 10 kg block is pulled with a force of 60 N on a rough surface with coefficient of kinetic friction 0.3. What is the acceleration of the block? (Take \( g = 9.8 \text{ m/s}^2 \))
A · 3.14 m/s²
Friction force \( f_k = \mu_k mg = 0.3 \times 10 \times 9.8 = 29.4 \text{ N} \). Net force = 60 - 29.4 = 30.6 N. Acceleration = 30.6/10 = 3.06 m/s² approx 3.14 m/s².
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Refer to the free body diagram below of a block on a slope with friction. If the block is stationary, which force balances the component of weight down the slope?
A · Static friction force
Static friction balances the component of weight parallel to the slope to keep the block stationary.
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A 15 kg box is pulled horizontally with a force of 100 N. The coefficient of kinetic friction is 0.2. What is the acceleration of the box?
A · 4.67 m/s²
Friction force = 0.2 × 15 × 9.8 = 29.4 N; Net force = 100 - 29.4 = 70.6 N; acceleration = 70.6 / 15 = 4.71 m/s² approx 4.67 m/s². Correct answer is A.
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Refer to the free body diagram below of a block on a frictionless incline of 45°. What is the acceleration of the block if mass is 2 kg?
A · \( 9.8 \times \sin 45^\circ \) m/s²
Acceleration down the incline is \( g \sin \theta = 9.8 \times \sin 45^\circ \approx 6.93 \text{ m/s}^2 \).
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Which of the following best defines angular momentum \( \vec{L} \) of a particle about a point O?
B · \( \vec{L} = \vec{r} \times \vec{p} \)
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Which statement correctly describes the physical meaning of angular momentum?
C · It quantifies the tendency of a particle to continue rotating about a point.
Angular momentum quantifies the tendency of a particle or system to maintain its rotational motion about a point or axis, analogous to linear momentum in translational motion.
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Refer to the diagram below showing a particle with position vector \( \vec{r} \) and linear momentum \( \vec{p} \). What is the direction of the angular momentum \( \vec{L} = \vec{r} \times \vec{p} \)?
C · Perpendicular to the plane containing \( \vec{r} \) and \( \vec{p} \) following right-hand rule
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Which of the following is the correct mathematical expression for the angular momentum \( \vec{L} \) of a rigid body rotating about a fixed axis?
B · \( \vec{L} = I \vec{\omega} \)
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A particle moves in the xy-plane with position vector \( \vec{r} = 3\hat{i} + 4\hat{j} \) m and linear momentum \( \vec{p} = 2\hat{i} + 5\hat{j} \) kg·m/s. What is the magnitude of its angular momentum about the origin?
A · 7 kg·m²/s
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Refer to the diagram below showing a figure skater spinning with arms extended. When the skater pulls in the arms, what happens to the angular momentum and angular velocity?
B · Angular momentum remains constant, angular velocity increases
In absence of external torque, angular momentum is conserved. When the skater pulls in the arms, moment of inertia decreases, so angular velocity increases to keep angular momentum constant.
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A spinning disk initially at rest is subjected to a constant torque \( \tau = 4\, \mathrm{N\cdot m} \). After 5 seconds, what is the change in its angular momentum?
A · \( 20\, \mathrm{kg\cdot m^2/s} \)
Change in angular momentum \( \Delta L = \tau \times t = 4 \times 5 = 20 \, \mathrm{kg\cdot m^2/s} \).
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Torque \( \vec{\tau} \) is related to angular momentum \( \vec{L} \) by which of the following expressions?
A · \( \vec{\tau} = \frac{d\vec{L}}{dt} \)
Torque is the time derivative of angular momentum, i.e., \( \vec{\tau} = \frac{d\vec{L}}{dt} \).
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Refer to the diagram below showing a force \( \vec{F} \) applied at a point on a rotating wheel at radius \( r \). Which expression correctly gives the torque \( \tau \) about the center?
B · \( \tau = F r \sin \theta \)
Torque magnitude is given by \( \tau = F r \sin \theta \), where \( \theta \) is the angle between \( \vec{F} \) and the position vector \( \vec{r} \).
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A solid disk of moment of inertia \( I = 2\, \mathrm{kg\cdot m^2} \) rotates with angular velocity \( \omega = 10\, \mathrm{rad/s} \). What is its angular momentum?
A · \( 20\, \mathrm{kg\cdot m^2/s} \)
Angular momentum \( L = I \omega = 2 \times 10 = 20 \, \mathrm{kg\cdot m^2/s} \).
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A rigid body rotates about a fixed axis with angular velocity \( \vec{\omega} \). Which of the following statements is true about its angular momentum \( \vec{L} \)?
A · \( \vec{L} \) is always parallel to \( \vec{\omega} \)
For rotation about a fixed axis, angular momentum \( \vec{L} = I \vec{\omega} \) is parallel to the angular velocity vector \( \vec{\omega} \).
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Which of the following is an application of angular momentum in planetary motion?
A · Planets move in elliptical orbits due to conservation of angular momentum
According to Kepler's second law, the conservation of angular momentum causes planets to sweep equal areas in equal times, resulting in elliptical orbits.
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Which of the following best describes the physical meaning of angular momentum \( \vec{L} \) in classical mechanics?
C · The quantity of rotation of a body, dependent on its moment of inertia and angular velocity
Angular momentum \( \vec{L} \) represents the quantity of rotation of a body and depends on both its moment of inertia and angular velocity, reflecting how much rotational motion the body has.
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Angular momentum is a vector quantity. Which of the following statements about its direction is correct?
B · It is perpendicular to the plane formed by the position and linear momentum vectors, following the right-hand rule
Angular momentum \( \vec{L} = \vec{r} \times \vec{p} \) is perpendicular to the plane formed by \( \vec{r} \) and \( \vec{p} \), with direction given by the right-hand rule.
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A particle of mass 2 kg moves in a circle of radius 0.5 m with a speed of 4 m/s. What is the magnitude of its angular momentum about the center of the circle?
A · 4 kg·m²/s
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Refer to the diagram below showing a uniform solid disk of mass 3 kg and radius 0.4 m rotating about its central axis at 10 rad/s. What is the angular momentum of the disk about its axis?
A · 2.4 kg·m²/s
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A figure skater spins with arms extended and has an angular momentum of 12 kg·m²/s. When she pulls her arms in, her moment of inertia decreases to half. What is her new angular velocity assuming no external torque?
C · Twice the original angular velocity
Conservation of angular momentum \( L = I\omega \) means \( I_1 \omega_1 = I_2 \omega_2 \). If \( I_2 = \frac{1}{2} I_1 \), then \( \omega_2 = 2 \omega_1 \).
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Refer to the diagram below showing a force \( \vec{F} \) applied at a point on a rigid body at a distance \( r \) from the axis of rotation. Which expression correctly gives the torque \( \tau \) about the axis?
A · \( \tau = F r \sin \theta \)
Torque is given by \( \tau = r F \sin \theta \), where \( \theta \) is the angle between \( \vec{r} \) and \( \vec{F} \).
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A constant torque \( \tau = 5 \) N·m acts on a rigid body initially at rest. After 4 seconds, the angular momentum of the body is:
A · 20 N·m·s
Torque \( \tau = \frac{dL}{dt} \) so \( L = \tau t = 5 \times 4 = 20 \) N·m·s.
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A particle moves under a central force such that its angular momentum vector changes direction but not magnitude. Which of the following statements is always true?
D · The angular momentum vector is constant in magnitude but not in direction due to external torque.
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A particle of mass 2.4 kg moves in a circle of radius 0.45 m with constant speed 5.3 m/s. What is the rate of change of its angular momentum about the center of the circle?
A · 0 N·m
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A particle of mass 0.9 kg moves in the xy-plane with velocity v = (4 i + 3 j) m/s and position vector r = (2 i + 5 j) m. What is the z-component of its angular momentum about the origin?
B · 3.15 kg·m²/s
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Which of the following best defines Simple Harmonic Motion (SHM)?
A · Motion where acceleration is proportional and opposite to displacement
SHM is defined as motion where the restoring force (and hence acceleration) is directly proportional to displacement and directed towards the mean position.
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In SHM, the acceleration \( a \) is related to displacement \( x \) by which equation?
A · \( a = -\omega^2 x \)
The acceleration in SHM is given by \( a = -\omega^2 x \), where \( \omega \) is the angular frequency and the negative sign indicates acceleration is directed opposite to displacement.
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Which characteristic is NOT true for Simple Harmonic Motion?
C · The velocity is maximum at maximum displacement
In SHM, velocity is zero at maximum displacement and maximum at the mean position, so option C is incorrect.
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A particle executes SHM with amplitude \( A \) and angular frequency \( \omega \). What is the expression for its displacement \( x(t) \) at time \( t \)?
A · \( x(t) = A \sin(\omega t + \phi) \)
The displacement in SHM is generally represented as \( x(t) = A \sin(\omega t + \phi) \) or \( A \cos(\omega t + \phi) \), where \( \phi \) is the phase constant.
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Refer to the diagram below showing displacement \( x \) versus time \( t \) for an SHM. What is the time period \( T \) of the oscillation?
A · The time between two successive maxima
The period \( T \) is the time interval between two successive maxima (or minima) in the displacement-time graph.
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Given the displacement equation of SHM as \( x = 0.05 \sin(100t) \) meters, what is the frequency of oscillation?
A · 15.9 Hz
Angular frequency \( \omega = 100 \) rad/s, frequency \( f = \frac{\omega}{2\pi} = \frac{100}{2\pi} \approx 15.9 \) Hz. Option D is closest to 15.9 Hz.
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Which of the following expressions correctly represents the velocity \( v \) of a particle executing SHM with amplitude \( A \) and angular frequency \( \omega \) at displacement \( x \)?
A · \( v = \pm \omega \sqrt{A^2 - x^2} \)
Velocity in SHM is given by \( v = \pm \omega \sqrt{A^2 - x^2} \), derived from energy conservation.
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A particle executes SHM with displacement \( x = A \cos(\omega t + \phi) \). Which of the following is the correct expression for acceleration \( a \)?
A · \( a = -\omega^2 A \cos(\omega t + \phi) \)
Acceleration is the second derivative of displacement, giving \( a = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x \).
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In SHM, what is the total mechanical energy \( E \) of the system with mass \( m \), amplitude \( A \), and angular frequency \( \omega \)?
A · \( E = \frac{1}{2} m \omega^2 A^2 \)
Total energy in SHM is constant and given by \( E = \frac{1}{2} m \omega^2 A^2 \).
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Refer to the energy-time graph below for a particle in SHM. Which curve represents kinetic energy \( K \)?
A · Curve that is maximum when displacement is zero
Kinetic energy is maximum when displacement is zero (mean position) and zero at maximum displacement.
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What is the potential energy \( U \) of a particle executing SHM at displacement \( x \) from the mean position?
A · \( U = \frac{1}{2} m \omega^2 x^2 \)
Potential energy in SHM is given by \( U = \frac{1}{2} m \omega^2 x^2 \), proportional to the square of displacement.
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If the total mechanical energy in SHM is \( E \), what is the kinetic energy \( K \) at displacement \( x \)?
A · \( K = E - \frac{1}{2} m \omega^2 x^2 \)
Kinetic energy at displacement \( x \) is total energy minus potential energy: \( K = E - U = E - \frac{1}{2} m \omega^2 x^2 \).
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A damped oscillator experiences a force proportional to velocity. Which term correctly represents this damping force?
A · \( F_d = -b v \)
Damping force is proportional and opposite to velocity: \( F_d = -b v \), where \( b \) is the damping coefficient.
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Refer to the graph below showing amplitude decay versus time for a damped oscillator. What type of damping is represented if amplitude decreases exponentially?
A · Under-damping
Exponential decay of amplitude indicates under-damping where oscillations continue with reducing amplitude.
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Which of the following conditions corresponds to critical damping in a damped harmonic oscillator?
A · \( b = 2 \sqrt{m k} \)
Critical damping occurs when the damping coefficient \( b = 2 \sqrt{m k} \), where \( m \) is mass and \( k \) is spring constant.
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In a damped oscillator, the logarithmic decrement \( \delta \) is defined as:
A · \( \delta = \frac{1}{n} \ln \frac{A_1}{A_{n+1}} \)
Logarithmic decrement \( \delta \) measures the rate of amplitude decay and is defined as \( \delta = \frac{1}{n} \ln \frac{A_1}{A_{n+1}} \).
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Which of the following describes over-damping in a damped harmonic oscillator?
A · The system returns to equilibrium without oscillating
In over-damping, the system returns to equilibrium slowly without oscillating.
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Refer to the diagram below of a forced oscillator. Which parameter represents the driving frequency?
A · Frequency of the external force \( \omega \)
The driving frequency \( \omega \) is the frequency of the external periodic force applied to the system.
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What happens to the amplitude of a forced oscillator when the driving frequency approaches the natural frequency?
A · Amplitude reaches maximum (resonance)
At resonance, when driving frequency equals natural frequency, amplitude reaches a maximum.
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Refer to the resonance curve below. What does the sharpness of the peak indicate?
A · Quality factor (Q) of the oscillator
The sharpness of the resonance peak is related to the quality factor \( Q \), indicating how underdamped the system is.
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Which of the following is true about the phase difference between the driving force and displacement at resonance in a forced oscillator?
A · Phase difference is \( \frac{\pi}{2} \)
At resonance, the displacement lags the driving force by \( \frac{\pi}{2} \) radians.
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The period of a simple pendulum depends on:
A · Length of the pendulum and acceleration due to gravity
Period \( T = 2\pi \sqrt{\frac{l}{g}} \) depends only on length \( l \) and gravity \( g \), independent of mass.
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Refer to the diagram of a simple pendulum below. Which angle represents the angular displacement \( \theta \)?
A · Angle between the string and vertical
Angular displacement \( \theta \) is the angle between the string and the vertical line.
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The moment of inertia \( I \) of a physical pendulum about the pivot is related to its period \( T \) by:
A · \( T = 2\pi \sqrt{\frac{I}{mgd}} \)
Period of physical pendulum is \( T = 2\pi \sqrt{\frac{I}{mgd}} \), where \( d \) is distance from pivot to center of mass.
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Which of the following statements is true for a physical pendulum compared to a simple pendulum?
A · Physical pendulum's period depends on its moment of inertia
Physical pendulum's period depends on moment of inertia and distance to center of mass, unlike simple pendulum which depends only on length.
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Two pendulums of equal length but different masses are set oscillating. Which statement is correct about their periods?
A · Both have the same period
Period of simple pendulum is independent of mass; both pendulums have the same period if length is equal.
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Two identical pendulums are coupled by a spring as shown in the diagram below. What phenomenon is observed when both pendulums oscillate with slightly different frequencies?
A · Beats
When two oscillations of close frequencies superpose, beats occur, producing periodic variation in amplitude.
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The beat frequency produced by two oscillations of frequencies \( f_1 \) and \( f_2 \) is given by:
A · \( |f_1 - f_2| \)
Beat frequency is the absolute difference between the two frequencies: \( |f_1 - f_2| \).
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Refer to the waveform diagram below showing beats formed by two waves of close frequencies. What does the envelope represent?
A · Amplitude variation of resultant wave
The envelope shows the amplitude modulation due to interference of two waves with close frequencies, producing beats.
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In coupled oscillators, normal modes refer to:
A · Specific patterns of oscillation where all parts oscillate with the same frequency
Normal modes are characteristic oscillations where the system oscillates at a single frequency with all parts moving in a fixed pattern.
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Which of the following affects the beat frequency produced by two coupled oscillators?
A · Difference in their natural frequencies
Beat frequency depends on the difference between the natural frequencies of the two oscillators.
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In a damped oscillator, the amplitude after one complete oscillation decreases from \( A_0 \) to \( A_1 \). The logarithmic decrement \( \delta \) is given by:
A · \( \delta = \ln \frac{A_0}{A_1} \)
Logarithmic decrement is the natural logarithm of the ratio of successive amplitudes: \( \delta = \ln \frac{A_0}{A_1} \).
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In forced oscillations, the amplitude \( A \) as a function of driving frequency \( \omega \) is given by:
A · \( A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (\frac{b\omega}{m})^2}} \)
Amplitude in forced oscillations depends on driving frequency, natural frequency, damping, and force amplitude as given by the formula.
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Which of the following best describes Simple Harmonic Motion (SHM)?
B · Motion where acceleration is directly proportional to displacement and opposite in direction
In SHM, acceleration \( a \) is proportional to displacement \( x \) but acts in the opposite direction, i.e., \( a = -\omega^2 x \).
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A particle executes SHM with amplitude \( A \). What is the displacement of the particle when its velocity is maximum?
A · Zero displacement
Velocity is maximum when displacement is zero because all energy is kinetic at equilibrium position.
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Which of the following quantities remains constant during Simple Harmonic Motion?
C · Total mechanical energy
Total mechanical energy (sum of kinetic and potential energy) remains constant in ideal SHM without damping.
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If the time period of a simple pendulum is \( T \), what will be the time period when the length of the pendulum is quadrupled?
A · \( 2T \)
Time period \( T = 2\pi \sqrt{\frac{l}{g}} \). If length \( l \) is quadrupled, \( T' = 2\pi \sqrt{\frac{4l}{g}} = 2T \).
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A particle moves in SHM with displacement \( x = A \sin(\omega t + \phi) \). Which of the following statements is true about the phase \( \phi \)?
B · It determines the initial position and velocity of the particle
Phase constant \( \phi \) sets the initial conditions (initial displacement and velocity) of the oscillation.
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The displacement of a particle executing SHM is given by \( x = 0.1 \sin(10t) \) meters. What is the maximum velocity of the particle?
A · 1 m/s
Maximum velocity \( v_{max} = \omega A = 10 \times 0.1 = 1 \) m/s.
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Which of the following is the correct expression for acceleration \( a \) in SHM if displacement is \( x = A \cos(\omega t) \)?
A · \( a = -\omega^2 A \cos(\omega t) \)
Acceleration is the second derivative of displacement: \( a = -\omega^2 x = -\omega^2 A \cos(\omega t) \).
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If the displacement of a particle in SHM is given by \( x = 0.05 \sin(20t + \frac{\pi}{3}) \), what is the phase constant?
A · \( \frac{\pi}{3} \)
The phase constant is the term added inside the sine function, here \( \frac{\pi}{3} \).
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Refer to the diagram below showing velocity \( v \) versus time \( t \) for a particle in SHM. What is the time period of oscillation?
B · 2 s
The velocity graph completes one full cycle in 2 seconds, so the time period \( T = 2 \) s.
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What is the expression for total energy \( E \) of a particle executing SHM with mass \( m \), angular frequency \( \omega \), and amplitude \( A \)?
A · \( E = \frac{1}{2} m \omega^2 A^2 \)
Total energy in SHM is constant and given by \( E = \frac{1}{2} m \omega^2 A^2 \).
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At what displacement \( x \) is the kinetic energy of a particle executing SHM equal to its potential energy?
A · \( x = \frac{A}{\sqrt{2}} \)
Kinetic energy equals potential energy when \( x = \frac{A}{\sqrt{2}} \).
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The total energy of a particle in SHM is 0.5 J and amplitude is 0.1 m. What is the maximum kinetic energy?
A · 0.5 J
Maximum kinetic energy equals total energy in SHM, which is 0.5 J here.
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Refer to the diagram below showing potential energy \( U \) and kinetic energy \( K \) versus displacement \( x \) for SHM. At which point is the kinetic energy maximum?
A · At \( x = 0 \)
Kinetic energy is maximum at equilibrium position \( x = 0 \) where potential energy is minimum.
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In a damped oscillator, which of the following quantities decreases exponentially with time?
A · Amplitude
In damped oscillations, amplitude decreases exponentially due to energy loss.
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Refer to the diagram below showing amplitude versus time for a damped oscillator. What type of damping is represented if the amplitude decreases slowly over many cycles?
A · Light damping
Light damping causes amplitude to decrease slowly over many oscillations.
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In a damped harmonic oscillator, the logarithmic decrement \( \delta \) is defined as:
A · \( \delta = \ln \frac{A_n}{A_{n+1}} \)
Logarithmic decrement \( \delta \) is the natural log of ratio of successive amplitudes.
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Which of the following statements is true for critical damping?
A · System returns to equilibrium without oscillating in minimum time
Critical damping returns the system to equilibrium as quickly as possible without oscillations.
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The amplitude of a damped oscillator decreases to half in 10 cycles. What is the logarithmic decrement \( \delta \)?
A · \( \delta = 0.0693 \)
\( \delta = \frac{1}{n} \ln \frac{A_0}{A_n} = \frac{1}{10} \ln 2 = 0.0693 \).
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In forced oscillations, resonance occurs when the driving frequency equals:
A · Natural frequency of the system
Resonance happens when the driving frequency matches the natural frequency, causing maximum amplitude.
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Refer to the diagram below showing amplitude versus driving frequency for a forced oscillator. What happens to the amplitude at resonance?
A · Amplitude reaches maximum
At resonance, amplitude peaks due to energy input matching natural frequency.
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Which of the following factors does NOT affect the resonance frequency of a forced oscillator?
C · Amplitude of driving force
Amplitude of driving force affects amplitude but not resonance frequency.
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In a forced oscillation system with damping, what happens to the resonance frequency compared to the natural frequency?
A · Resonance frequency is slightly less than natural frequency
Damping lowers the resonance frequency slightly below the natural frequency.
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The moment of inertia of a physical pendulum about its pivot is \( I \) and distance from pivot to center of mass is \( d \). What is the expression for its time period \( T \)?
A · \( T = 2\pi \sqrt{\frac{I}{mgd}} \)
Time period of physical pendulum is \( T = 2\pi \sqrt{\frac{I}{mgd}} \).
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Refer to the diagram below of a simple pendulum. If the length is \( l \) and mass is \( m \), what is the restoring torque when displaced by angle \( \theta \)?
A · \( -mg l \sin \theta \)
Restoring torque is \( \tau = -mg l \sin \theta \), acting opposite to displacement.
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Which of the following expressions gives the time period \( T \) of a simple pendulum of length \( l \) in gravitational field \( g \)?
A · \( T = 2\pi \sqrt{\frac{l}{g}} \)
Time period of simple pendulum is \( T = 2\pi \sqrt{\frac{l}{g}} \).
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If the length of a simple pendulum is doubled, what happens to its frequency?
A · Frequency decreases by factor \( \frac{1}{\sqrt{2}} \)
Frequency \( f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{l}} \), so doubling \( l \) reduces frequency by \( \frac{1}{\sqrt{2}} \).
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The time period of a mass-spring system is \( T \). If the mass is increased by 4 times and spring constant is doubled, what is the new time period?
A · \( T \sqrt{2} \)
\( T = 2\pi \sqrt{\frac{m}{k}} \). New \( T' = 2\pi \sqrt{\frac{4m}{2k}} = 2\pi \sqrt{2 \frac{m}{k}} = T \sqrt{2} \).
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Refer to the diagram below of a mass attached to a spring. If the spring constant is \( k \) and mass is \( m \), what is the angular frequency \( \omega \) of oscillations?
A · \( \sqrt{\frac{k}{m}} \)
Angular frequency for mass-spring system is \( \omega = \sqrt{\frac{k}{m}} \).
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What is the phase difference between displacement and velocity in SHM?
A · \( \frac{\pi}{2} \)
Velocity leads displacement by \( \frac{\pi}{2} \) in phase.
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If two oscillations have phase difference \( \pi \), what is the nature of their motion?
A · They are in antiphase (opposite phase)
Phase difference of \( \pi \) means the oscillations are exactly out of phase (antiphase).
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Refer to the diagram below showing two SHM displacement-time graphs with a phase difference. What is the phase difference between the two oscillations?
B · \( \frac{\pi}{2} \)
The second graph lags the first by a quarter period, corresponding to \( \frac{\pi}{2} \) phase difference.
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Two oscillations of the same frequency and amplitude are superposed with a phase difference of \( \frac{\pi}{2} \). What is the resultant amplitude?
A · \( A \sqrt{2} \)
Resultant amplitude \( = 2A \cos \frac{\phi}{2} = 2A \cos \frac{\pi}{4} = A \sqrt{2} \).
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Which of the following oscillators is commonly used in clocks due to its accurate frequency?
A · Tuning fork
Tuning forks have stable frequency and are widely used in clocks and timekeeping.
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Which property of a tuning fork makes it suitable for use in musical instruments?
A · Produces nearly pure tone with fixed frequency
Tuning forks produce nearly pure tones with fixed frequency, ideal for musical tuning.
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A clock pendulum is shortened by 1%. How does this affect the clock's timekeeping?
A · Clock runs faster
Shortening pendulum decreases period, so clock runs faster.
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A damped harmonic oscillator has mass m, spring constant k, and damping coefficient b. If the system is critically damped, which of the following relations between m, k, and b holds true?
A · A) b = 2 √(m k)

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