CH 14 SIMPLE HARMONIC MOTION PRACTICE TEST SOLUTION BY PHYSICS INN

 

PRACTICE TEST PAPER CHAPTER 14: SIMPLE HARMONIC MOTION

ANSWER KEY

VERSION 1 — LOW DIFFICULTY (SOLUTIONS)

SECTION I: MULTIPLE CHOICE QUESTIONS

1. (b) Vibration
• Reason/Explanation: By definition, one complete round trip (cycle) of a vibrating body about its mean position is called a vibration or oscillation.
2. (c) Hertz
• Reason/Explanation: The SI unit of frequency is hertz (), which is equal to one oscillation per second ().
3. (b) Mean position
• Reason/Explanation: S.H.M is a special type of oscillatory motion where the restoring force always pulls or directs the object back towards its mean (equilibrium) position.
4. (c) Amplitude
• Reason/Explanation: The magnitude of the maximum displacement of the vibrating body on either side of its mean position is defined as amplitude ().
5. (b) Second pendulum
• Reason/Explanation: A pendulum that completes one vibration in two seconds (i.e., time period ) is known as a second pendulum.
6. (b)
• Reason/Explanation: Frequency  is mathematically expressed as the reciprocal of the time period  ().
7. (c)
• Reason/Explanation: According to Hooke's law, the elastic restoring force exerted by a spring is , where the negative sign indicates it is opposite to displacement.
8. (c) Maximum
• Reason/Explanation: At the mean position (), all energy is kinetic, making the velocity maximum ().
9. (c) Damped oscillations
• Reason/Explanation: Oscillations with decreasing amplitude over time in the presence of resistive forces (friction, air drag) are called damped oscillations.
10. (b) Resonance
• Reason/Explanation: A microwave oven uses high-frequency electromagnetic waves to excite water molecules at their natural frequency, heating food through resonance.
11. (b) Elastic potential energy
• Reason/Explanation: The work done against the restoring force in stretching/compressing a spring is stored in the system as elastic potential energy.
12. (b) Antinodes
• Reason/Explanation: High flames occur where large amounts of gas escape at high-pressure regions (antinodes) of the acoustic standing wave.
13. (a) Small
• Reason/Explanation: The formula for S.H.M of a simple pendulum is derived assuming a small angle/amplitude ().

SECTION II: SHORT QUESTIONS

1. Oscillatory Motion & Examples:
• Definition: Oscillatory (or vibratory) motion is a to-and-fro periodic motion of a body about a fixed mean position, repeating itself after equal intervals of time.
• Examples: Motion of a simple pendulum, motion of a child's swing, or vibration of guitar strings.
2. Frequency vs. Angular Frequency:
• Frequency (): The number of complete vibrations/cycles completed by an oscillating body per second (). Measured in Hertz ().
• Angular Frequency (): The rate of change of angular displacement per unit time (). Measured in radians per second ().
3. Hooke's Law:
• Hooke's Law states that within elastic limits, the applied force (or restoring force) is directly proportional to the displacement produced from the mean position.
• Formula: , where  is the spring constant.
4. Second Pendulum:
• Definition: A simple pendulum whose time period of oscillation is exactly 2 seconds ().
• Time Period: .
5. Restoring Force:
• Definition: A force that always acts on an oscillating body in a direction pointing back toward its equilibrium/mean position, attempting to restore it to rest.
6. Decrease in Swing Amplitude:
• The amplitude decreases over time because dissipating forces such as air resistance and mechanical friction continuously convert the mechanical energy of the swing into heat/thermal energy.
7. Free Oscillations:
• Definition: Oscillations executed by a body moving with its natural frequency without the influence or interference of any external periodic driving force.
• Example: A simple pendulum slightly displaced and left to swing freely.
8. Role of Shock Absorbers in Vehicles:
• Shock absorbers damp unwanted oscillations when a car travels over bumps. They dissipate kinetic energy using liquid damping, bringing the vehicle body quickly back to equilibrium to ensure passenger comfort.
9. Phase in S.H.M:
• Phase is the angle  (or ) that specifies both the instantaneous displacement  and the direction of motion of the oscillator at any given time .
10. Condition for Resonance:
• Resonance occurs when the driving frequency of an external periodic force matches the natural frequency of the oscillating system ().

VERSION 2 — MEDIUM DIFFICULTY (SOLUTIONS)

SECTION I: MULTIPLE CHOICE QUESTIONS

1. (c)  times
• Reason/Explanation: Since , . Doubling the length () makes .
2. (b)
• Reason/Explanation: Displacement is represented as a cosine function () and velocity as a sine function (). The phase difference between them is  radians ().
3. (c)  (in radians)
• Reason/Explanation: For small angular displacements (),  when  is expressed in radians.
4. (b) Direction is towards mean position
• Reason/Explanation: In , the negative sign mathematically indicates that acceleration  is always directed opposite to displacement , toward the mean position.
5. (c) Velocity
• Reason/Explanation: At extreme positions (), the object momentarily stops to reverse direction, making instantaneous velocity zero ().
6. (c) Square of amplitude
• Reason/Explanation: Total energy equation is . Thus, .
7. (b) Shortest possible time
• Reason/Explanation: Critical damping allows a system to return to its equilibrium position in the shortest possible time without overshooting/oscillating.
8. (b) Sharpness of resonance
• Reason/Explanation: Radio tuning relies on matching electrical circuit natural frequency to incoming signals via resonance to absorb maximum energy cleanly.
9. (b)
• Reason/Explanation: Frequency .
10. (b) Lead has higher inertia
• Reason/Explanation: Lead has a much higher mass/density, giving it greater inertia to resist air drag compared to a lightweight pith-ball.
11. (b) Out of phase
• Reason/Explanation: When phase difference is  ( radians), the oscillators move in opposite directions at any instant and are said to be  out of phase.
12. (b)
• Reason/Explanation: Comparing S.H.M acceleration  with  gives .
13. (b) Standing wave patterns
• Reason/Explanation: Chladni plates form nodal line patterns with sprinkled sand when driven at resonant frequencies, demonstrating 2D standing waves.

SECTION II: SHORT QUESTIONS

1. Small Amplitude for Simple Pendulum:
• The equation of S.H.M for a pendulum uses the approximation . This holds true only for small angles (). Large amplitudes cause motion to become non-harmonic.
2. Light Damping vs. Heavy Damping:
• Light Damping: Small resistive forces where energy and amplitude decay gradually over many cycles (e.g., swing in air).
• Heavy Damping: Very strong drag where the system takes a long time to return to rest without completing regular oscillations (e.g., pendulum in thick oil).
3. Pendulum Time Period on Moon:
• Formula: . Since gravity on the Moon () is approximately  of Earth's gravity (), the smaller  increases the time period .
4. Effect of Mass on Mass-Spring System:
• Formula: . Time period is directly proportional to the square root of mass (). Adding more mass increases .
5. Angular Frequency ():
• Angular displacement per unit time (). Its SI unit is .
6. Soldiers Breaking Steps on Bridges:
• Marching in step provides a periodic force. If the marching frequency equals the bridge's natural frequency, resonance occurs, causing high-amplitude vibrations that could destroy the bridge.
7. Point where :
• Equating : . The energies are equal at a displacement of .
8. Forced Oscillations:
• Oscillations produced when an oscillating body is driven continuously by an external periodic force rather than vibrating naturally. Example: A swing being continuously pushed.
9. Resonance vs. Damping Sharpness:
• Sharpness of resonance depends on damping. Smaller damping results in a sharper, higher peak. Heavier damping flattens the resonance curve.
10. Pendulum at Center of Earth:
• At the center of the Earth, gravitational acceleration . Since ,  and restoring force . Thus, it cannot oscillate.

VERSION 3 — HIGH DIFFICULTY (SOLUTIONS)

SECTION I: MULTIPLE CHOICE QUESTIONS

1. (b)
• Reason/Explanation: Horizontal projection velocity of circular motion gives S.H.M velocity equation .
2. (b) Decrease
• Reason/Explanation: Standing up raises the child's center of mass, shortening effective pendulum length . Since , time period  decreases.
3. (b) Initial displacement and velocity
• Reason/Explanation: The initial phase constant  in  is determined entirely by the initial position  and velocity  at time .
4. (c) Extreme position
• Reason/Explanation: Acceleration is . Magnitude  is maximum when displacement  is maximum ().
5. (c) Critical
• Reason/Explanation: Critical damping brings the displaced system back to rest in the minimum time interval without oscillatory overshoot.
6. (b) 4 times
• Reason/Explanation: Total energy . Since , doubling frequency quadruples energy ().
7. (b) Nodes
• Reason/Explanation: At nodes, acoustic pressure variations create minimal sound pressure regions where particles remain trapped against gravity.
8. (b)
• Reason/Explanation: In mass-spring systems, , which means .
9. (c)
• Reason/Explanation: Acceleration . The negative sign denotes a  ( rad) phase shift relative to displacement.
10. (c) Mass of the bob
• Reason/Explanation: Pendulum period  depends only on length  and gravitational acceleration , making it completely independent of mass .
11. (b) Atomic nuclei
• Reason/Explanation: MRI uses strong radio-frequency pulses that resonate with atomic nuclei (such as hydrogen protons in body tissue) to build images.
12. (b) Acoustic standing waves
• Reason/Explanation: Flame heights vary across top holes based on pressure nodes and antinodes formed by standing sound waves inside the gas-filled pipe.
13. (c)
• Reason/Explanation: For a second pendulum, period . Frequency .

SECTION II: SHORT QUESTIONS

1. EKG Needle Motion is Not S.H.M:
• An Electrocardiogram (EKG) traces periodic electrical pulses from heartbeats. However, its restoring force is not directly proportional to displacement () at all times. Thus, it is periodic motion, but not S.H.M.
2. Derivation of :
• By definition, angular velocity is .
• For one complete revolution, angle swept  and time taken .
• .
• Substituting  gives .
3. Determining Tower Height using Pendulum:
• Suspend a long wire bob from the tower top to act as a simple pendulum. Measure its oscillation period  with a stopwatch. Using , square both sides to solve for length :

• The calculated length  gives the height of the tower.
3. Phase Difference Between  and :
• Velocity: .
• Acceleration: .
• Phase difference:  ().
4. Physical Significance of Phase:
• Phase  describes the exact state of motion of an oscillator at any given time . It specifies both the instant position (displacement ) and the direction of motion.
5. Working of Acoustic Levitation:
• Sound waves from an ultrasonic transducer reflect off a surface, forming acoustic standing waves. Pressure differences between high-pressure antinodes and low-pressure nodes create a net acoustic radiation force that supports small objects against gravity at the nodes.
6. Damping Effect on Sharpness of Resonance:
• Heavy damping flattens the resonance curve, decreasing maximum amplitude and broadening response bandwidth. Light damping produces a sharp peak with high resonant amplitude.
7. Flame Heights in Rubens Tube:
• Acoustic standing waves inside the tube create pressure variations. At pressure antinodes, high pressure forces more gas out, producing tall flames. At pressure nodes, low pressure lets less gas escape, producing short flames.
8. Pendulum Clock at Mount Everest:
• At Mount Everest, distance from the Earth's center is greater, so gravitational acceleration  decreases. Since , a smaller  increases the time period . As a result, the pendulum swings slower, causing the clock to lose time (run slow).
9. Calculate Angular Frequency ( for ):
• Given:
• Formula:
• Calculation: