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Physics M.C.Q (1 Marks)

Simple Harmonic Motion · Rajasthan Board (RBSE) STD 12 Science (Rajasthan - English Medium). 37 questions.

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M.C.Q (1 Marks)

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37 questions · 7 auto-graded MCQ + 30 self-marked written.

Question 11 Mark
A particle moves in the X - Y plane according to the equation $\overrightarrow{\text{r}}=\Big(\overrightarrow{\text{i}}+2\overrightarrow{\text{j}}\Big)\text{A}\cos\omega\text{t}.$ The motion of the particle is:
  1. On a straight line.
  2. On an ellipse.
  3. Periodic.
  4. Simple harmonic.
Answer
  1. on a straight line.
  1. Periodic.
  1. Simple harmonic.
Explanation:
The given equation is a solution to the equation of simple harmonic motion. The amplitude is $\big(\overrightarrow{\text{i}}+2\overrightarrow{\text{j}}\big)\text{A},$ following equation of straight line y = mx + c. Also, a simple harmonic motion is periodic.
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MCQ 21 Mark
A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is $T$. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will:
  • A
    Remain $T$
  • B
    Become $2T$
  • C
    Become $\frac{\text{T}}{2}$
  • ✓
    Become $\frac{\text{T}}{\sqrt{2}}$
Answer
Correct option: D.
Become $\frac{\text{T}}{\sqrt{2}}$
Time period $(T)$ is given by,
$\text{T}=2\pi\sqrt{\frac{\text{m}}{\text{k}}}$
where $m$ is the mass, and $k$ is spring constant.
When the spring is divided into two parts, the new spring constant $k_1$ is given as,
$\text{k}_1=2\text{k}$
New time period $T_1$ :
$\text{T}_1=2\pi\sqrt{\frac{\text{m}}{2\text{k}}}=\frac{1}{\sqrt{2}}2\pi\sqrt{\frac{\text{m}}{\text{k}}}=\frac{1}{\sqrt{2}}\text{T}$
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MCQ 31 Mark
The displacement of a particle is given by $\overrightarrow{\text{r}}=\text{A}\big(\overrightarrow{\text{i}}\cos\omega\text{t}+\overrightarrow{\text{j}}\sin\omega\text{t}\big)$ The motion of the particle is$:$
  • A
    Simple harmonic.
  • B
    On a straight line.
  • ✓
    On a circle.
  • D
    With constant acceleration.
Answer
Correct option: C.
On a circle.
We know, $\frac{\text{d}^2}{\text{dt}^2}\overrightarrow{\text{r}}=-\omega^2\overrightarrow{\text{r}}$
But there is a phase difference of $90^{\circ}$ between the $x$ and $y$ components because of which the particle executes a circular motion and hence, the projection of the particle on the diameter executes a simple harmonic motion.
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MCQ 41 Mark
Two bodies $A$ and $B$ of equal mass are suspended from two separate massless springs of spring constant $k_1$ and $k_2$ respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of $A$ to that of $B$ is:
  • A
    $\frac{\text{k}_1}{\text{k}_2}$
  • B
    $\sqrt{\frac{\text{k}_1}{\text{k}_2}}$
  • C
    $\frac{\text{k}_2}{\text{k}_1}$
  • ✓
    $\sqrt{\frac{\text{k}_2}{\text{k}^1}}$
Answer
Correct option: D.
$\sqrt{\frac{\text{k}_2}{\text{k}^1}}$
Maximum velocity, $\text{v}=\text{A}\omega$
where $A$ is amplitude and $\omega$ is the angular frequency.
Further, $\omega=\sqrt{\frac{\text{k}}{\text{m}}}$
Let $A$ and $B$ be the amplitudes of particles $A$ and $B$ respectively.
As the maximum velocity of particles are equal,
i.e. $v_A = v_B$
or,
$\text{A}\omega_\text{A}=\text{B}\omega_\text{B}$
$\Rightarrow\text{A}\sqrt{\frac{\text{k}_1}{\text{m}_\text{A}}}=\text{B}\sqrt{\frac{\text{k}_2}{\text{m}_\text{B}}}$
$\Rightarrow\text{A}\sqrt{\frac{\text{k}_1}{\text{m}}}=\text{B}\sqrt{\frac{\text{k}_2}{\text{m}}}\big(\text{m}_\text{A}=\text{m}_{\text{B}}=\text{m}\big)$
$\Rightarrow\frac{\text{A}}{\text{B}}=\sqrt{\frac{\text{k}_2}{\text{k}_1}}$
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Question 51 Mark
A particle moves in a circular path with a uniform speed. Its motion is:
  1. Periodic.
  2. Oscillatory.
  3. Simple harmonic.
  4. Angular simple harmonic.
Answer
  1. Periodic.
Explanation:
Because the particle covers one rotation after a fixed interval of time but does not oscillate around a mean position.
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Question 61 Mark
A particle executes simple harmonic motion with a frequency v. The frequency with which the kinetic energy oscillates is:
  1. $\frac{\text{v}}{2}$
  2. v
  3. 2v
  4. zero
Answer
  1. 2v
Explanation:
Because in one complete oscillation, the kinetic energy changes its value from zero to maximum, twice.
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Question 71 Mark
The motion of a torsional pendulum is:
  1. Periodic.
  2. Oscillatory.
  3. Simple harmonic.
  4. Angular simple harmonic.
Answer
  1. Periodic.
  1. Oscillatory.
  1. Angular simple harmonic.
Explanation:
Because it completes one oscillation in a fixed interval of time and the oscillations are in terms of rotation of the body through some angle.
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Question 81 Mark
Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is:
  1. Simple harmonic.
  2. Parabolic.
  3. On a straight line.
  4. Periodic.
Answer
  1. On a straight line.
  2. Periodic.
Explanation:
If the particle were dropped from the surface of the earth, the motion of the particle would be SHM. But when it is dropped from a height h, the motion of the particle is not SHM because there is no horizontal velocity imparted. In that case, the motion of the particle would be periodic and in a straight line.
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Question 91 Mark
The time period of a particle in simple harmonic motion is equal to the time between consecutive appearances of the particle at a particular point in its motion. This point is:
  1. The mean position.
  2. An extreme position.
  3. Between the mean position and the positive extreme.
  4. Between the mean position and the negative extreme.
Answer
  1. An extreme position.
Explanation:
One oscillation is said to be completed when the particle returns to the extreme position i.e. from where it started.
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Question 101 Mark
An object is released from rest. The time it takes to fall through a distance h and the speed of the object as it falls through this distance are measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated:
  1. The measured times are same.
  2. The measured speeds are same.
  3. The actual times in the fall are equal.
  4. The actual speeds are equal.
Answer
  1. The measured times are same.
  2. The measured speeds are same.
Explanation:
The effect of gravity on the object as well as on the pendulum clock is same in both cases the time measured is also same. As the time measured is same, the speeds are same.
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MCQ 111 Mark
The time period of a particle in simple harmonic motion is equal to the smallest time between the particle acquiring a particular velocity $\overrightarrow{\text{v}}.$ The value of $v$ is
  • A
    $v_{max}.$
  • B
    $0$
  • ✓
    between $0$ and $v_{max}.$
  • D
    between $0$ and $- v_{max}$
Answer
Correct option: C.
between $0$ and $v_{max}.$
Because the time period of a simple harmonic motion is defined as the time taken to complete one oscillation
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Question 121 Mark
A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is:
  1. Periodic.
  2. Oscillatory.
  3. Simple harmonic.
  4. Angular simple harmonic.
Answer
  1. Periodic.
Explanation:
Because the particle completes one rotation in a fixed interval of time but does not oscillate around a mean position.
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Question 131 Mark
The distance moved by a particle in simple harmonic motion in one time period is:
  1. A
  2. 2A
  3. 4A
  4. zero.
Answer
  1. 4A
Explanation:
In an oscillation, the particle goes from one extreme position to other extreme position that lies on the other side of mean position and then returns back to the initial extreme position. Thus, total distance moved by particle is,
2A + 2A = 4A.
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Question 141 Mark
A wall clock uses a vertical spring-mass system to measure the time. Each time the mass reaches an extreme position, the clock advances by a second. The clock gives correct time at the equator. If the clock is taken to the poles it will:
  1. Run slow.
  2. Run fast.
  3. Stop working.
  4. Give correct time.
Answer
  1. Give correct time.
Explanation:
Because the time period of a spring-mass system does not depend on the acceleration due to gravity.
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Question 151 Mark
Figure represents two simple harmonic motions. The parameter which has different values in the two motions is:
  1. Amplitude.
  2. Frequency.
  3. Phase.
  4. Maximum velocity.
Answer
  1. Phase.
Explanation:
Because the direction of motion of particles A and B is just opposite to each other.
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Question 161 Mark
A spring-mass system occillates in a car. if the car accelerates on a horizontal road, the frequency of oscillation will:
  1. Increase.
  2. Decrease.
  3. Remain same.
  4. Become zero.
Answer
  1. Remain same.
Explanation:
As the frequency of the system is independent of the acceleration of the system.
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Question 171 Mark
Select the correct statements:
  1. A simple harmonic motion is necessarily periodic.
  2. A simple harmonic motion is necessarily oscillatory.
  3. An oscillatory motion is necessarily periodic.
  4. A periodic motion is necessarily oscillatory.
Answer
  1. A simple harmonic motion is necessarily periodic.
  2. A simple harmonic motion is necessarily oscillatory.
Explanation:
A periodic motion need not be necessarily oscillatory.
For example: The moon revolving around the earth.
Also, an oscillatory motion need not be necessarily periodic.
For example: Damped harmonic motion.
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Question 181 Mark
The displacement of a particle in simple harmonic motion in one time period is:
  1. A
  2. 2A
  3. 4A
  4. zero.
Answer
  1. zero.
Explanation:
Displacement is defined as the distance between the starting and the end point through a straight line. In one complete oscillation, the net displacement is zero as the particle returns to its initial position.
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Question 191 Mark
The free end of a simple pendulum is attached to the ceiling of a box. The box is taken to a height and the pendulum is oscillated. When the bob is at its lowest point, the box is released to fall freely. As seen from the box during this period, the bob will:
  1. Continue its oscillation as before.
  2. Stop.
  3. Will go in a circular path.
  4. Move on a straight line.
Answer
  1. Will go in a circular path.
Explanation:
As the acceleration due to gravity acting on the bob of pendulum, due to free fall gives a torque to the pendulum, the bob goes in a circular path.
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Question 201 Mark
A particle moves in a circular path with a continuously increasing speed. Its motion is:
  1. Periodic.
  2. Oscillatory.
  3. Simple harmonic.
  4. None of them.
Answer
  1. None of them.
Explanation:
As the particle does not complete one rotation in a fixed interval of time, neither does it oscillate around a mean position.
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Question 211 Mark
The motion of a particle is given by $\text{x = A}\sin\omega\text{t+B}\cos\omega\text{t}.$ The motion of the particle is:
  1. Not simple harmonic.
  2. Simple harmonic with amplitude A + B.
  3. Simple harmonic with amplitude $\frac{\text{(A}+\text{B)}}{2}.$
  4. Simple harmonic with amplitude $\sqrt{(\text{A}^2+\text{B}^2)}.$
Answer
  1. Simple harmonic with amplitude $\sqrt{(\text{A}^2+\text{B}^2)}.$
Explanation:
$\text{x}=\text{A}\sin\omega\text{t}+\text{B}\cos\omega\text{t}\ ...(1)$
Acceleration,
$\text{a}=\frac{\text{d}^2\text{x}}{\text{d}\text{t}^2}=\frac{\text{d}^2}{\text{dt}^2}(\text{A}\sin\omega\text{t}+\text{B}\cos\omega\text{t})$
$=\frac{\text{d}}{\text{dt}}(\text{A}\omega\cos\omega\text{t}-\text{B}\omega\sin\omega\text{t)}$
$=-\text{A}\omega^2\sin\omega\text{t}=\text{B}\omega^2\cos\omega\text{t}$
$=-\omega^2(\text{A}\sin\omega\text{t}+\text{B}\cos\omega\text{t})$
$=-\omega^2\text{x}$
For a body to undergo simple harmonic motion,
acceleration, a = -kx .....(2)
Therefore, from the equations (1) and (2), it can be seen that the given body undergoes simple harmonic motion with amplitude, $\text{A}=\sqrt{\text{A}^2+\text{B}^2}.$
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Question 221 Mark
Which of the following will change the time period as they are taken to moon?
  1. A simple pendulum.
  2. A physical pendulum.
  3. A torsional pendulum.
  4. A spring-mass system.
Answer
  1. A simple pendulum.
  2. A physical pendulum.
Explanation:
As the time period of a simple pendulum and a physical pendulum depends on the acceleration due the gravity, the time period of these pendulums changes when they are taken to the moon.
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MCQ 231 Mark
The total mechanical energy of a spring$-$mass system in $1$ simple harmonic motion is $\text{E}=\frac{1}{2}\text{m}\omega^2\text{A}^2.$ Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude A remains the same. The new mechanical energy will:
  • A
    Become $2E$
  • B
    Become $\frac{\text{E}}{2}$
  • C
    Become $\sqrt{2\text{E}}$
  • ✓
    Remain $E.$
Answer
Correct option: D.
Remain $E.$
Mechanical energy $(E)$ of a spring$-$mass system in simple harmonic motion is given by,
$\text{E}=\frac{1}{2}\text{m}\omega^2\text{A}^2$
where $m$ is mass of body, and $\omega$ is angular frequency.
Let $m_1$ be the mass of the other particle and $\omega_1$ be its angular frequency.
New angular frequency $\omega_1$ is given by,
$\omega_1=\sqrt{\frac{\text{k}}{\text{m}_1}}=\sqrt{\frac{\text{k}}{2\text{m}}}\big(\text{m}_1=2\text{m}\big)$
New energy $E_1​​​​​​​$is given as,
$\text{E}_1=\frac{1}{2}\text{m}_1\omega_1^2\text{A}^2$
$=\frac{1}{2}\big(2\text{m}\big)\Big(\sqrt{\frac{\text{k}}{2\text{m}}}\Big)^2\text{A}^2$
$=\frac{1}{2}\text{m}\omega^2\text{A}^2=\text{E}$
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Question 241 Mark
The average energy in one time period in simple harmonic motion is:
  1. $\frac{1}{2}\text{m}\omega^2\text{A}^2$
  2. $\frac{1}{4}\text{m}\omega^2\text{A}^2$
  3. $\text{m}\omega^2\text{A}^2$
  4. $\text{Zero}$
Answer
  1. $\frac{1}{2}\text{m}\omega^2\text{A}^2$
Explanation:
It is the total energy in simple harmonic motion in one time period.
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Question 251 Mark
The average acceleration in one time period in a simple harmonic motion is:
  1. $\text{A}\omega^2$
  2. $\frac{\text{A}\omega^2}{2}$
  3. $\frac{\text{A}\omega^2}{\sqrt{2}}$
  4. $\text{Zero}.$
Answer
  1. $\text{Zero}.$
Explanation:
The acceleration changes its direction (to opposite direction) after every half oscillation. Thus, net acceleration is given as,
$\text{A}\omega^2+(-\text{A}\omega^2)=0$
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MCQ 261 Mark
A particle moves on the $X-$axis according to the equation $\text{x = A + B}\sin\omega\text{t}.$ The motion is simple harmonic with amplitude:
  • A
    $A$
  • ✓
    $B$
  • C
    $A + B$
  • D
    $\sqrt{\text{A}^2+\text{B}^2}$
Answer
Correct option: B.
$B$
At $t = 0,$
Displacement $(x_0)$ is given by,
$\text{x}_0=\text{A}+\sin\omega(0)=\text{A}$
Displacement $x$ will be maximum when $\sin\omega\text{t}$ is $1$
or,
$x_m = A + B$
Amplitude will be:
$x_m- x_0 = A + B - A = B$
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Question 271 Mark
A student says that he had applied a force $\text{F}=-\text{k}\sqrt{\text{x}}$ on a particle and the particle moved in simple harmonic motion. He refuses to tell whether k is a constant or not. Assume that he was worked only with positive x and no other force acted on the particle.
  1. As x increases k increases.
  2. As x increases k decreases.
  3. As x increases k remains constant.
  4. The motion cannot be simple harmonic.
Answer
  1. As x increases k increases.
Explanation:
A body is said to be in simple harmonic motion only when,
F = -kx .....(1)
where F is force,
k is force constant, and,
x is displacement of the body from the mean position.
Given:
$\text{F}=-\text{k}\sqrt{\text{x}}\ ...(2)$

On comparing the equations (1) and (2), it can be said that in order to execute simple harmonic motion, k should be proportional to $\sqrt{\text{x}}$ Thus, as x increases k increases.
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Question 281 Mark
Which of the following quantities are always negative in a simple harmonic motion?
  1. $\overrightarrow{\text{F}}.\overrightarrow{\text{a}}$
  2. $\overrightarrow{\text{v}}.\overrightarrow{\text{r}}$
  3. $\overrightarrow{\text{a}}.\overrightarrow{\text{r}}$
  4. $\overrightarrow{\text{F}}.\overrightarrow{\text{r}}$
Answer
  1. $\overrightarrow{\text{a}}.\overrightarrow{\text{r}}$
  2. $\overrightarrow{\text{F}}.\overrightarrow{\text{r}}$
Explanation:
In S.H.M.,
F = -kx
Therefore, $\overrightarrow{\text{F}}.\overrightarrow{\text{r}}$ will always be negative. As acceleration has the same direction as the force, $\overrightarrow{\text{a}}.\overrightarrow{\text{r}}$ will also be negative, always.
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Question 291 Mark
Which of the following quantities are always zero in a simple harmonic motion?
  1. $\overrightarrow{\text{F}}\times\overrightarrow{\text{a}}$
  2. $\overrightarrow{\text{v}}\times\overrightarrow{\text{r}}$
  3. $\overrightarrow{\text{a}}\times\overrightarrow{\text{r}}$
  4. $\overrightarrow{\text{F}}\times\overrightarrow{\text{r}}$
Answer
  1. $\overrightarrow{\text{F}}\times\overrightarrow{\text{a}}$
  2. $\overrightarrow{\text{v}}\times\overrightarrow{\text{r}}$
  3. $\overrightarrow{\text{a}}\times\overrightarrow{\text{r}}$
  4. $\overrightarrow{\text{F}}\times\overrightarrow{\text{r}}$
Explanation:
As $\overrightarrow{\text{F}},\overrightarrow{\text{a}},\overrightarrow{\text{r}}$ and $\overrightarrow{\text{v}}$ are either parallel or anti-parallel to each other, their cross products will always be zero.
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MCQ 301 Mark
A particle moves on the $X-$axis according to the equation $\text{x}=\text{x}_0\sin^2\omega\text{t.}$ The motion is simple harmonic.
  • A
    With amplitude $x_0$
  • B
    With amplitude $2x_0$
  • C
    With time period $\frac{2\pi}{\omega}$.
  • ✓
    With time period $\frac{\pi}{\omega}.$
Answer
Correct option: D.
With time period $\frac{\pi}{\omega}.$
Given equation:
$\text{x}=\text{x}_0\sin^2\omega\text{t}$
$\Rightarrow\text{x}=\frac{\text{x}_0}{2}\big(\cos2\omega\text{t}-1\big)$
Now, the amplitude of the particle is $=\frac{\text{x}_0}{2}$ and the angular frequency of the $\text{SHM}$ is $2\omega.$
Thus, time period of the $\text{SHM} \frac{2\pi}{\text{angular frequecy}}=\frac{2\pi}{2\omega}=\frac{\pi}{\omega}$
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Question 311 Mark
Which of the following quantities are always positive in a simple harmonic motion?
  1. $\overrightarrow{\text{F}}.\overrightarrow{\text{a}}$
  2. $\overrightarrow{\text{v}}.\overrightarrow{\text{r}}$
  3. $\overrightarrow{\text{a}}.\overrightarrow{\text{r}}$
  4. $\overrightarrow{\text{F}}.\overrightarrow{\text{r}}$
Answer
  1. $\overrightarrow{\text{F}}.\overrightarrow{\text{a}}$
Explanation:
As the direction of force and acceleration are always same, $\overrightarrow{\text{F}}.\overrightarrow{\text{a}}.$ is always positive.
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Question 321 Mark
A pendulum clock keeping correct time is taken to high altitudes:
  1. It will keep correct time.
  2. Its length should be increased to keep correct time.
  3. Its length should be decreased to keep correct time.
  4. It cannot keep correct time even if the length is changed.
Answer
  1. Its length should be decreased to keep correct time.
Explanation:
Time period of pendulum,
$\text{T}=2\pi\sqrt{\frac{\text{l}}{\text{g}}}$
At higher altitudes, the value of acceleration due to gravity decreases.
Therefore, the length of the pendulum should be decreased to compensate for the decrease in the value of acceleration due to gravity.
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Question 331 Mark
A pendulum clock that keeps correct time on the earth is taken to the moon. It will run:
  1. At correct rate.
  2. 6 times faster.
  3. $\sqrt{6}$ times faster.
  4. $\sqrt{6}$ times slower.
Answer
  1. $\sqrt{6}$ times slower.
Explanation:
The acceleration due to gravity at moon is $\frac{\text{g}}{6}. $
Time period of pendulum is given by, $\text{T}=2\pi\sqrt{\frac{\text{l}}{g}}$
Therefore, on moon, time period will be:
$\text{T}_{\text{moon}}=2\pi\sqrt{\frac{\text{l}}{\text{g}_{\text{moon}}}}=2\pi\sqrt{\frac{\text{l}}{\Big(\frac{\text{g}}{6}\Big)}}$
$=\sqrt{6}\Big(2\pi\sqrt{\frac{\text{l}}{\text{g}}}\Big)=\sqrt{6}\text{T}$
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Question 341 Mark
In a simple harmonic motion:
  1. The maximum potential energy equals the maximum kinetic energy.
  2. The minimum potential energy equals the minimum kinetic energy.
  3. The minimum potential energy equals the maximum kinetic energy.
  4. The maximum potential energy equals the minimum kinetic energy.
Answer
  1. The maximum potential energy equals the maximum kinetic energy.
  2. The minimum potential energy equals the minimum kinetic energy.
Explanation:
In SHM,
maximum kinetic energy $=\frac{1}{2}\text{kA}^2$
maximum potential energy $=\frac{1}{2}\text{kA}^2$
The minimum value of both kinetic and potential energy is zero. 
Therefore, in a simple harmonic motion the maximum kinetic energy and maximum potential energy are equal. Also, the minimum kinetic energy and the minimum potential energy are equal.
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Question 351 Mark
For a particle executing simple harmonic motion, the acceleration is proportional to:
  1. Displacement from the mean position.
  2. Distance from the mean position.
  3. Distance travelled since t = 0.
  4. Speed.
Answer
  1. Displacement from the mean position.
Explanation:
For S.H.M.,
F = -kx
ma = -kx (F = ma)
or,
$\text{a}=-\frac{\text{k}}{\text{m}}\text{x}$
Thus, acceleration is proportional to the displacement from the mean position but in opposite direction.
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Question 361 Mark
A spring-mass system oscillates with a frequency v. If it is taken in an elevator slowly accelerating upward, the frequency will:
  1. Increase.
  2. Decrease.
  3. Remain same.
  4. Become zero.
Answer
  1. Remain same.
Explanation:
Because the frequency $\Big(\text{v}-\frac{1}{2\pi}\sqrt{\frac{\text{k}}{\text{m}}}\Big)$ of the system is independent of the acceleration of the system.
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Question 371 Mark
In a simple harmonic motion:
  1. The potential energy is always equal to the kinetic energy.
  2. The potential energy is never equal to the kinetic energy.
  3. The average potential energy in any time interval is equal to the average kinetic energy in that time interval.
  4. The average potential energy in one time period is equal to the average kinetic energy in this period.
Answer
  1. The average potential energy in one time period is equal to the average kinetic energy in this period.
Explanation:
The kinetic energy of the motion is given as,
$\frac{1}{2}\text{K}\text{A}^2\cos^2\omega\text{t}$
The potential energy is calculated as,
$\frac{1}{2}\text{K}\text{A}^2\sin^2\omega\text{t}$
As the average of the cosine and the sine function is equal to each other over the total time period of the functions, the average potential energy in one time period is equal to the average kinetic energy in this period.
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