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All questions of Periodic Motion (PHY) for MCAT Exam

Find the amplitude of the S.H.M whose displacement y in cm is given by equation y= 3sin 157t +4cos157t where t is time in seconds.
  • a)
    20Hz
  • b)
    25Hz
  • c)
    50Hz
  • d)
    40Hz
Correct answer is option 'B'. Can you explain this answer?

When the displacement of a SHM is:
y=a sin wt+ b cos wt
  • Amplitude of the SHM will be:
    A=√a2+b2
Here, a = 3, b = 4
Amplitude, A= √(32+42) = 5 cm
Hence option B is correct.

A second pendulum is mounted in a space shuttle. Its period of oscillations will decrease when rocket is
  • a)
    moving in geostationary orbit
  • b)
    ascending up with uniform acceleration
  • c)
    descending down with uniform acceleration
  • d)
    moving up with uniform velocity
Correct answer is option 'B'. Can you explain this answer?

Top Rankers answered
 
  • Time Period, T = 2π √(l/g')where,
    l = Length of seconds pendulum 
    g’ = Apparent Gravity
  • For the period of oscillations of Seconds Pendulum to decrease, the Apparent gravity (g’) has to increase because:
  • Hence, Time Period of oscillations of Seconds Pendulum will decrease when the rocket is ascending up with uniform acceleration.

If a simple pendulum oscillates with an amplitude 50 mm and time period 2s, then its maximum velocity is
a)0.15 m/s
b)0.1 m/s
c)0.16 m/s
d)0.8 m/s
Correct answer is option 'A'. Can you explain this answer?

Neha Joshi answered
We know that in a simple harmonic motion the maximum velocity,
Vmax = A⍵
Here A = 50 mm
And ⍵ = 2π / T
= 2π / 2
= π
Hence  Vmax = 50 x 10-3
= 0.15 m/s

What will be the phase difference between bigger pendulum (with time period 5T/4 )and smaller pendulum (with time period T) after one oscillation of bigger pendulum?
  • a)
    π/4
  • b)
    π/2
  • c)
    π/3
  • d)
    π
Correct answer is option 'B'. Can you explain this answer?

Preeti Iyer answered
By the time bigger pendulum completes one vibration, the smaller pendulum would have completed 5/4 vibrations. That is smaller pendulum will be ahead by 1/4 vibration in phase. 1/4 vibration means λ/4 path or π/2 radians.

A frequency of 1Hz corresponds to:
  • a)
    2 vibrations per second
  • b)
    1 vibration per second
  • c)
    10 vibrations per second
  • d)
    a time period of ½ second
Correct answer is option 'B'. Can you explain this answer?

Alok Mehta answered
Frequency used to be measured in cycles per second, but now we use the unit of frequency - the Hertz (abbreviated Hz). One Hertz (1Hz) is equal to one vibration per second. So the weight above is bouncing with a frequency of about 1Hz. The sound wave corresponding to Middle C on a piano is around 256Hz.

A rubber ball with water, having a small hole in its bottom is used as the bob of a simple pendulum. The time-period of such a pendulum:
  • a)
    Increases with time
  • b)
    First increases and then decreases finally having same value as at the beginning
  • c)
    Decreases with time
  • d)
    Is a constant
Correct answer is option 'B'. Can you explain this answer?

Rajesh Gupta answered
When rubber ball completely filled with water its centre of gravity will be at its centre, as water will fall through hole its COG will shift towards lower side leading to increase in length of pendulum and thus T, when very small amount of water will be left in rubber ball its COG will again shifts upward causing decrease in length and thus T, and finally when rubber ball becomes empty its COG will be at its centre and T will remains same as earlier.

Damped natural frequency is
  • a)
    same as natural frequency
  • b)
    lower than natural frequency
  • c)
    higher than natural frequency
  • d)
    none of the above
Correct answer is option 'B'. Can you explain this answer?

Geetika Shah answered
If a resonant mechanical structure is set in motion and left to its own devices, it will continue to oscillate at a particular frequency known as its natural frequency, or "damped natural frequency". This will be a little lower in frequency than the resonant frequency, which is the frequency it would assume if there were no damping. The resonant frequency is also called the "undamped natural frequency”

In a simple pendulum the restoring force is due to
  • a)
     The tangential component of the gravitational force
  • b)
    The radial component of the gravitational force
  • c)
    The tangential component of the tension in string
  • d)
    The radial component of the tension in string
Correct answer is option 'A'. Can you explain this answer?

Pooja Shah answered
In a simple pendulum the restoring force is due to the tangential component of the gravitational force because on applying torque equation on the radial forces the effect is nullified due to it passing from centre & only tangential component remains.

The kinetic energy of a body executing S.H.M. is 1/3 of the potential energy. Then, the displacement of the body is x percent of the amplitude, where x is
  • a)
    33
  • b)
    50
  • c)
    67
  • d)
    87
Correct answer is option 'A'. Can you explain this answer?

Pooja Shah answered
We know that PE + KE = TE = constant
Hence at the extremis TE = PE = ½ kz2  
Where z is amplitude and k is shm constant.
Thus when KE = ⅓ PE
We get PE = ¾ TE =  ½ kz2
Hence we get  ½ kx2 = ¾  ½ kz2
We get x/z = √3/4 
= 1.73 / 2
= .87
Thus we get x is 87 percent of the amplitude.

The pendulum of a wall clock executes
  • a)
    a translatory motion
  • b)
    a rotary motion
  • c)
    a random motion
  • d)
    an oscillatory motion
Correct answer is option 'D'. Can you explain this answer?

Chirag Mehta answered
The bob of a pendulum moves in such a way that it repeats its positions several time but alternately the time gap between these positions is always equal which proves that the motion of a pendulum is oscillatory.

If the maximum velocity and acceleration of a particle executing S.H.M are equal in magnitude, the time period will be
  • a)
    6.28 s
  • b)
    1.57 s
  • c)
    3.14 s
  • d)
    12.56 s
Correct answer is option 'A'. Can you explain this answer?

Jyoti Dey answered
**Explanation:**

When a particle is executing Simple Harmonic Motion (SHM), its displacement from the mean position follows a sinusoidal function. The maximum displacement of the particle is called the amplitude (A) of the motion. The particle oscillates back and forth between two extreme positions, which are equal in magnitude but opposite in direction.

The time taken for one complete oscillation of the particle is called the time period (T) of the motion. It is the time taken for the particle to go from one extreme position to the other and back again.

The maximum velocity (vmax) of the particle occurs when the particle passes through the mean position (equilibrium position). At this point, the velocity is at a maximum and is in the same direction as the acceleration. The maximum acceleration (amax) of the particle occurs when the particle is at its extreme positions. At these points, the acceleration is at a maximum and is in the opposite direction to the displacement.

If the maximum velocity and acceleration of the particle are equal in magnitude, it means that the amplitude of the motion is equal to the acceleration. Mathematically, this can be represented as:

amax = vmax

Using the equations of SHM, we can relate the amplitude, time period, maximum velocity, and maximum acceleration as follows:

amax = ω^2 A

vmax = ω A

where ω is the angular frequency of the motion.

Since amax = vmax, we can equate the above two equations:

ω^2 A = ω A

Simplifying the equation:

ω^2 = ω

ω = 1

The angular frequency ω is related to the time period T by ω = 2π/T. Substituting this value into the equation:

(2π/T)^2 = 2π/T

Simplifying the equation:

4π^2/T^2 = 2π/T

2π/T = 4π^2/T^2

T = 2

Therefore, the time period of the motion is T = 2 seconds, which is equivalent to 6.28 seconds (approximately). Hence, the correct answer is option A: 6.28 s.

A body is executing S.H.M When its displacement from the mean position is 4 cm and 5 cm, the corresponding velocities of the body are 10 cm per sec and 8 cm per sec. Then the time period of the body is
  • a)
    2 π/2 sec
  • b)
    π sec
  • c)
    3 π/2 sec
  • d)
    2 π sec
Correct answer is option 'B'. Can you explain this answer?

Pranavi Chopra answered
v=ω√A2−x2
​⇒8=ω √A2−25​
&10=ω√A2−16
​⇒(8/10​)2= A2−25/ A2−16
​⇒16/25​= A2−25/ A2−16
​⇒16A2−256=25A2−625
⇒9A2=369
A2=41
⇒8=ω41−25
​⇒8=4ω
∴ω=2=T2π​
T=π sec​

A particle executes SHM of time period T, the time period with which the total energy changes is
  • a)
    2T
  • b)
    T
  • c)
    T/2
  • d)
    Infinite
Correct answer is option 'D'. Can you explain this answer?

Kavita Joshi answered
Potential energy is a function of height. Now the motion of a simple pendulum can be expressed in terms of the angle at which the string makes with the normal (the angle it would hang at if it didn't swing.)Now Because the bob of the pendulum is always in motion except at the two turning points where the angle is at a maximum (and even at these points the bob is only motionless at one instant in time but for no inteval) the hight is always changing for every inteval of time and thus the potential energy is always changing.Also the potential energy is the same for all values of the angle that have the same magnatude (independent of the sign.)

The periodic time (tp) is given by
  • a)
    ω / 2 π
  • b)
    2 π / ω
  • c)
    2 π × ω
  • d)
    π/ω
Correct answer is option 'B'. Can you explain this answer?

Divyansh Saha answered
Periodic time is the time taken for one complete revolution of the particle.
∴ Periodic time, tp = 2 π/ω seconds.

For all practical purposes, the motion of a simple pendulum is SHM,
  • a)
    None of these
  • b)
    Only if the length of its string is at least one meter
  • c)
    Only if the maximum angle which its string makes with the vertical is less than 342.
  • d)
    Only if the maximum angle which the string makes with the vertical is less than I
Correct answer is option 'A'. Can you explain this answer?

Nitin Nair answered
The motion of Simple Pendulum as Simple Harmonic Motion. When we pull a simple pendulum from its equilibrium position and then release it, it swings in a vertical plane under the influence of gravity. It begins to oscillate about its mean position. Therefore, the motion is periodic and oscillatory.
SO OPTION A IS CORRECT.

Period of a periodic motion is the ________ interval of time after which the motion is __________.
  • a)
    smallest, repeated
  • b)
    mean, not repeated
  • c)
    largest, repeated
  • d)
    average, repeated
Correct answer is option 'A'. Can you explain this answer?

Puja Das answered
Explanation:Time period of an oscillatory motion is the smallest interval of time for one complete oscillation.As when we apply the force on the bob,bob moves from mean position to extreme position then form extreme position to mean position and then again from mean to extreme in opposite direction and at last come to mean position.The time taken by bob in this whole process is called period of oscillation.

In simple models damping force is
  • a)
    directly proportional to velocity
  • b)
    inversely proportional to acceleration
  • c)
    inversely proportional to velocity
  • d)
    directly proportional to acceleration
Correct answer is option 'A'. Can you explain this answer?

Kritika Khanna answered
Damping force is a resistance force that opposes the motion of an object in a medium, such as air or water. It is responsible for reducing the amplitude of oscillations or the speed of a moving object. In simple models, the damping force is directly proportional to velocity.

Explanation:
The damping force is caused by the interaction between the object and the medium it is moving through. As the object moves, it displaces the medium particles, causing them to collide and exert a resistance force on the object. This resistance force is proportional to the velocity of the object.

Mathematical Representation:
The mathematical representation of damping force can be given by the equation:
F_damping = -bv

Where F_damping is the damping force, b is the damping coefficient, and v is the velocity of the object.

Directly Proportional to Velocity:
According to the equation, the damping force is directly proportional to the velocity of the object. This means that as the velocity increases, the damping force also increases, resulting in a greater resistance to the object's motion. Conversely, if the velocity decreases, the damping force decreases as well.

Relationship with Acceleration:
Although the damping force is not directly proportional to acceleration, it does have an indirect relationship with it. Since acceleration is the rate of change of velocity, an increase in acceleration results in a greater change in velocity over time. Therefore, a larger acceleration would lead to a larger damping force, and vice versa.

Conclusion:
In simple models, the damping force is directly proportional to velocity. This relationship is important in understanding the behavior of objects in the presence of damping forces, as it helps explain the gradual decrease in amplitude or speed of oscillations or motion.

The period of a simple pendulum is doubled when
  • a)
    The mass of the bob and the length of the pendulum are doubled
  • b)
    The mass of the bob is doubled
  • c)
    Its length is increased four times
  • d)
    Its length is doubled
Correct answer is option 'C'. Can you explain this answer?

The Period of a Simple Pendulum
The period of a simple pendulum is determined by the formula:
T = 2π√(L/g)
where:
- T is the period
- L is the length of the pendulum
- g is the acceleration due to gravity
Understanding the Influence of Length on Period
- The period T is directly proportional to the square root of the length L.
- This means if you increase the length L, the period T increases.
Explaining the Correct Answer (Option C)
- If the length of the pendulum is increased four times (L becomes 4L), the new period T' can be calculated as follows:
T' = 2π√(4L/g) = 2π√(4)√(L/g) = 2 * 2π√(L/g) = 2T
- This shows that the period T' is now doubled.
Analysis of Other Options
- Option A: Doubling mass and length does not double the period. The mass does not impact the period; it only affects the amplitude.
- Option B: Doubling the mass of the bob does not change the period at all; it remains T.
- Option D: Doubling the length (L becomes 2L) results in:
T' = 2π√(2L/g) = √2 * T (not doubled).
Conclusion
Thus, the only scenario where the period of a simple pendulum is doubled is when its length is increased four times, making Option C the correct answer.

Periodic motion is
  • a)
    motion that involves coming to a mean position at random intervals of time
  • b)
    motion that repeats itself at regular intervals of time
  • c)
    motion that does not repeat itself at regular intervals of time
  • d)
    motion that repeats itself at irregular intervals of time
Correct answer is option 'B'. Can you explain this answer?

Rajdeep Singh answered
Periodic motion is a type of motion that repeats itself at regular intervals of time. It is characterized by the recurring pattern of the motion, where the object or system returns to its initial state after a certain period of time. This motion can be observed in various natural and man-made phenomena, such as the swinging of a pendulum, the rotation of the Earth around its axis, the vibration of a guitar string, or the oscillation of a spring.

Characteristics of periodic motion:

- Repetition: Periodic motion involves the repetition of a certain pattern or motion over time. The object or system returns to its initial position, velocity, and other properties after completing one cycle of motion.

- Regular intervals: The time taken to complete one cycle of motion is constant and is known as the period. This means that the motion occurs at regular intervals of time. For example, the period of a pendulum is the time taken to swing from one extreme position to the other and back again.

- Consistent amplitude: In periodic motion, the amplitude remains constant throughout the motion. The amplitude represents the maximum displacement or distance from the mean position.

- Harmonic motion: Many periodic motions, such as the swinging of a pendulum or the vibration of a guitar string, can be described by simple harmonic motion. Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from the mean position and acts in the opposite direction.

- Periodic function: The mathematical representation of periodic motion is often described by a periodic function, such as a sine or cosine function. These functions repeat themselves after a certain interval, corresponding to one cycle of the motion.

- Examples of periodic motion: Some common examples of periodic motion include the swinging of a pendulum, the rotation of the Earth around its axis, the vibration of a guitar string, the oscillation of a spring, the motion of a bouncing ball, and the orbit of planets around the sun.

In conclusion, periodic motion is a type of motion that repeats itself at regular intervals of time. It is characterized by the recurring pattern of the motion, where the object or system returns to its initial state after completing one cycle of motion.

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