Question 511 Mark
How will the time period of a simple pendulum change when its length is doubled?
Answer
View full question & answer→It becomes $\sqrt{2}$ times the original time period.
Questions · Page 2 of 3
1 Marks Question
Question 521 Mark
Sometimes, when an automobile picks up speed, its body begins to rattle. Why?
Answer
View full question & answer→ This is because of resonant vibrations.
Question 531 Mark
If the displacement is represented by $\text{x}=3\sin\omega\text{t}+4\cos\omega\text{t},$ what is the amplitude?
Answer
View full question & answer→ Phase difference between $3\sin\omega\text{t}$ and $4\cos\omega\text{t}$ is $\frac{\pi}{2}.$$\therefore$ The amplitude is $\sqrt{3^2+4^2},$ i.e., 5 units.
Question 541 Mark
Is the damping force constant on a system executing SHM?
Answer
View full question & answer→No, because damping force depends upon velocity and is more when the system moves fast and is less when the system moves slow.
Question 551 Mark
Show that for a particle executing S.H.M, velocity and displacement have a phase difference of $\frac{\pi}{2}.$
Answer
View full question & answer→ Let the displacement equation of SHM$\text{x}=\text{a}\cos\omega\text{t}$
Velocity $\text{v}=\frac{\text{dx}}{\text{dt}}=\text{a}\omega(-\sin\omega\text{t})=-\text{a}\omega\sin\omega\text{t}$$\Rightarrow\text{v}=\text{a}\omega\cos\Big(\frac{\pi}{2}+\omega\text{t}\Big)$
Now, phase of displacement $\phi_1=\omega\text{t}$ Phase of velocity $\phi_2=\frac{\pi}{2}+\omega\text{t}$$\therefore$ Difference in phase of velocity to that of phase of displacement
$\triangle\phi=\phi_2-\phi_1=\Big(\frac{\pi}{2}+\omega\text{t}\Big)-(\omega\text{t})=\frac{\pi}{2}$
Velocity $\text{v}=\frac{\text{dx}}{\text{dt}}=\text{a}\omega(-\sin\omega\text{t})=-\text{a}\omega\sin\omega\text{t}$$\Rightarrow\text{v}=\text{a}\omega\cos\Big(\frac{\pi}{2}+\omega\text{t}\Big)$
Now, phase of displacement $\phi_1=\omega\text{t}$ Phase of velocity $\phi_2=\frac{\pi}{2}+\omega\text{t}$$\therefore$ Difference in phase of velocity to that of phase of displacement
$\triangle\phi=\phi_2-\phi_1=\Big(\frac{\pi}{2}+\omega\text{t}\Big)-(\omega\text{t})=\frac{\pi}{2}$
Question 561 Mark
Define force constant.
Answer
View full question & answer→Force constant is defined as the restoring force developed in a body per unit displacement.
Question 571 Mark
On what factors does the energy of a harmonic oscillator depend?
Answer
View full question & answer→Energy of a harmonic oscillator depends on the mass, frequency and amplitude of oscillation.
Question 581 Mark
What is the maximum value of the kinetic energy/ in the case of S.H.M.?
Answer
View full question & answer→Maximum value of K.E. is total energy, i.e. $\frac{1}{2}\text{m}\omega^2\text{A}^2.$
Question 591 Mark
Give the name of three important characteristics of a SHM.
Answer
View full question & answer→Three important characteristics of an SHM are amplitude, time period (or frequency) and phase.
Question 601 Mark
What provides restoring force in the following cases?
- A spring compressed and then left free to vibrate.
- Water disturbed in $U-$tube.
- Pendulum disturbed from its mean position.
Answer
View full question & answer→- Elasticity of the material of the spring.
- Weight of water.
- Weight of pendulum.
Question 611 Mark
When is the tension maximum in the string of a simple pendulum?
Answer
View full question & answer→At the lower-most point or mean position.
Question 621 Mark
What will be the time period of oscillation, if the length of a second pendulum is one third?
Answer
View full question & answer→$\frac{\text{T}_2^2}{\text{T}_2^1}=\frac{\text{l}_2}{\text{l}_1}=\frac{\text{l}}{\frac{3}{1}}=\frac{1}{3}$$\text{T}^2_2=\frac{(2)^2}{3}$
$\text{T}_2=\frac{2}{\sqrt{3}}\text{ sec}$
$\text{T}_2=\frac{2}{\sqrt{3}}\text{ sec}$
Question 631 Mark
Can a motion be periodic but not oscillatory? If your answer is yes, give an example and if not explain why?
Answer
View full question & answer→Yes, e.g., circular motion is periodic but not oscillatory.
Question 641 Mark
The amplitude of a harmonic oscillator is doubled. How does its energy change?
Answer
View full question & answer→As $\text{E}\propto\text{A}^2$ the energy of harmonic oscillator will became 4 times, its original value when its amplitude is doubled.
Question 651 Mark
Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? The rotation of earth about its axis.
Answer
View full question & answer→It is periodic but not simple harmonic motion because it is not to and fro about a fixed point.
Question 661 Mark
A simple harmonic motion is described by a = -16x where a → acceleration, x → displacement in m. What is the time period?
Answer
View full question & answer→ For S.H.M., $\text{a}=-\omega^2\text{x}$ Comparing with $\text{a}=-16\text{x}$$\because\omega^2=16\Rightarrow\omega=\frac{2\pi}{\text{T}}=\sqrt{16}=4$
$\therefore\text{T}=\frac{\pi}{2}\text{ second}$
$\therefore\text{T}=\frac{\pi}{2}\text{ second}$
Question 671 Mark
State force law for a simple harmonic motion.
Answer
View full question & answer→ Force $\text{F}\propto-\text{x}$$\Rightarrow\text{F = kx}$
$\Rightarrow\text{F}=-\text{m}\omega^2\text{x}.$
$\Rightarrow\text{F}=-\text{m}\omega^2\text{x}.$
Question 681 Mark
Is oscillation of a mass suspended by a spring simple harmonic in nature?
Answer
View full question & answer→Yes, it is if the spring is perfectly elastic.
Question 691 Mark
All oscillatory motions are periodic and vice-versa. Is it true?
Answer
View full question & answer→No, There are other types of periodic motions also. Circular motion and rotatory motion are periodic but non-oscillatory.
Question 701 Mark
What provides the restoring force for simple harmonic oscillations in the following cases?
- Simple pendulum.
- Spring.
- Column of mercury in $U-$tube.
Answer
View full question & answer→- Part of the force of gravity.
- Elastic restoring force.
- Force due to difference in column of mercury or pressure difference between the levels on the two limbs.
Question 711 Mark
What is the maximum velocity of the oscillating body?
Answer
View full question & answer→Maximum velocity is $\omega\text{A}.$
Question 721 Mark
A simple pendulum of length l suspended from a roof of a trolley which moves in a horizontal direction with an acceleration a. Find its time period of oscillation.
Answer
View full question & answer→The trolley is accelerated horizontally by a. So, there will be two accelerations, g vertically down and horizontal acceleration a. The net acceleration is $\sqrt{\text{g}^2+\text{a}^2}.$ The time period.$\text{T}=2\pi\sqrt{\frac{\text{l}}{\sqrt{\text{g}^2+\text{a}^2}}}.$
Question 731 Mark
What will be the change in the time period of a loaded spring when taken to moon?
Answer
View full question & answer→ No change, since $\text{T}=2\pi\sqrt{\frac{\text{m}}{\text{k}}}$
Question 741 Mark
A cylindrical wooden block of cross-section $15.0 \mathrm{~cm}^2$ and mass 230 gm is floated over water with an extra weight 50 gm attached to its bottom. The cylinder floats vertically. From the state of equilibrium, it is slightly depressed and released. If the specific gravity of wood is 0.30 and $\mathrm{g}=9.8 \mathrm{~m}$ per $\mathrm{sec}^2$, find the frequency of oscillation of the block.
Answer
View full question & answer→Area of cross-section of the block $=\pi \mathrm{r}^2=15 \mathrm{~cm}^2=15 \times 10^{-4} \mathrm{~m}^2$ Total weight of the block $=(230+50)=280 \mathrm{gm}=$ 0.28 kg Density of wood $=0.30 \mathrm{gm} / \mathrm{c} . \mathrm{c}=300 \mathrm{~kg} / \mathrm{m}^3$ Density of water $=10^3 \mathrm{~kg}$
When the cylinder is depressed in water through a distance $y$ the restoring force $=$ weight of water displaced $\mathrm{F}=$ Aydg $=\left(15 \times 10^{-4}\right) \times 10^3 \times 9.8$ newton/metre $=1.5 \times 9.8 \mathrm{~N} / \mathrm{m}$ Hence the frequenry of orillation is given by
$=\frac{1}{2\pi}\sqrt{\Big(\frac{\text{k}}{\text{m}}\Big)}$
$=\frac{1}{2\pi}\sqrt{\frac{1.5\times9.8}{0.28}}$
$=1.15\text{Hz}$
When the cylinder is depressed in water through a distance $y$ the restoring force $=$ weight of water displaced $\mathrm{F}=$ Aydg $=\left(15 \times 10^{-4}\right) \times 10^3 \times 9.8$ newton/metre $=1.5 \times 9.8 \mathrm{~N} / \mathrm{m}$ Hence the frequenry of orillation is given by
$=\frac{1}{2\pi}\sqrt{\Big(\frac{\text{k}}{\text{m}}\Big)}$
$=\frac{1}{2\pi}\sqrt{\frac{1.5\times9.8}{0.28}}$
$=1.15\text{Hz}$
Question 751 Mark
A particle is in linear simple harmonic motion between two points, A and B, 10cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is: At the end B.
Answer
From above figure, where A and B represent the two extreme positions of a SHM. For velocity, the direction from A to B is taken to b positive. The acceleration and the force, along AP are taken as positive and along Bp are taken as negative.
At the end B, velocity is zero. Here, acceleration and force are negative as they are directed along BP.
View full question & answer→From above figure, where A and B represent the two extreme positions of a SHM. For velocity, the direction from A to B is taken to b positive. The acceleration and the force, along AP are taken as positive and along Bp are taken as negative.At the end B, velocity is zero. Here, acceleration and force are negative as they are directed along BP.
Question 761 Mark
The maximum acceleration is a and the maximum velocity is v. What is the amplitude?
Answer
View full question & answer→ $\text{a}_{\text{max}}=-\omega^2\text{A = a, v}_{\text{max}}=\omega\text{A = v}$$\frac{\text{a}_{\text{max}}}{\text{v}_{\text{max}}}=\frac{\text{a}}{\text{v}}=-\omega,$
Also $\text{v}=\omega\text{A}=\frac{\text{a}}{\text{v}}\text{A}$
$\therefore\text{A}=\frac{\text{v}^2}{\text{a}}$
Also $\text{v}=\omega\text{A}=\frac{\text{a}}{\text{v}}\text{A}$
$\therefore\text{A}=\frac{\text{v}^2}{\text{a}}$
Question 771 Mark
What is the time period of oscillation of a pendulum in a satellite?
Answer
View full question & answer→Infinite.
Question 781 Mark
If the total energy with an oscillating system is E, what is the kinetic energy at $\text{x}=\frac{\text{A}}{3}?$
Answer
View full question & answer→K.E. $=\frac{1}{2}\text{m}\omega^2(\text{A}^2-\text{x}^2)$ Total energy, $\text{E}=\frac{1}{2}\text{m}\omega^2\text{A}^2$ At, $\text{x}=\frac{\text{A}}{3},$ K.E. $=\frac{1}{2}\text{m}\omega^2\Big(\text{A}^2-\frac{\text{A}^2}{9}\Big)$$=\frac{8}{9}\frac{1}{2}\text{m}\omega^2\text{A}^2=\frac{8}{9}\text{E}$
Question 791 Mark
What is I second's pendulum? What is the length of a second's pendulum?
Answer
View full question & answer→A pendulum having a time period of 2 seconds is called second's pendulum. Since $\text{T}=2\pi\sqrt{\frac{\text{l}}{\text{g}}},$$\text{l}=\frac{\text{T}^2\text{g}}{4\pi^2}=\frac{4\text{g}}{4\pi^2}=1\text{m}$
Question 801 Mark
Two simple pendulum of equal length cross each other at mean position. What is their phase difference?
Answer
View full question & answer→ $\pi$ radians.
Question 811 Mark
What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?
Answer
View full question & answer→f = 0.
Question 821 Mark
What is the frequency of variation of potential or kinetic energy when the frequency of the oscillation is f?
Answer
View full question & answer→Since K.E. or P.E. $\propto\cos^2\omega\text{t}$ or $\sin^2\omega\text{t},$ the frequency of variation is 2f.
Question 831 Mark
Two clocks, one working with oscillating pendulum and the other with spring are given. Which one will you use in satellite?
Answer
View full question & answer→The clock with spring is preferred, since it is not influenced by variation in g.
Question 841 Mark
Why the amplitude of the vibrating pendulum should be small?
Answer
View full question & answer→ For S.H.M., restoring force should always be pointing towards the mean position which is not possible at large angles.
Question 851 Mark
Which of the following conditions is not sufficient for $\text{S.H.M}$. and why?
- Acceleration displacement.
- Restoring force o displacement.
Answer
View full question & answer→Acceleration $\propto$ displacement is not sufficient since it does not refer the direction of these quantities. As you know acceleration is always against displacement.
Question 861 Mark
What determines the natural frequency of a body?
Answer
View full question & answer→Natural frequency of a body depends upon:
- Elastic properties of the material of the body and.
- Dimensions of the body.
Question 871 Mark
State the conditions when motion of a particle can be an SHM.
Answer
View full question & answer→For SHM, the restoring force on the particle must be proportional to its displacement and directed towards mean position.
Question 881 Mark
A particle is in linear simple harmonic motion between two points, A and B, 10cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is:At 3cm away from A going towards B.
Answer
From above figure, where A and B represent the two extreme positions of a SHM. For velocity, the direction from A to B is taken to b positive. The acceleration and the force, along AP are taken as positive and along Bp are taken as negative.
At 3cm away from A going towards B, the particle is at R, with a tendency to move along RP, which is positive direction. Here, velocity, acceleration all are positive.
View full question & answer→From above figure, where A and B represent the two extreme positions of a SHM. For velocity, the direction from A to B is taken to b positive. The acceleration and the force, along AP are taken as positive and along Bp are taken as negative.At 3cm away from A going towards B, the particle is at R, with a tendency to move along RP, which is positive direction. Here, velocity, acceleration all are positive.
Question 891 Mark
Can a simple pendulum vibrate at the centre of Earth?
Answer
View full question & answer→No. This is because of zero value of g at the centre of Earth.
Question 901 Mark
A simple pendulum is inside a space craft. What should be its time period of vibration?
Answer
View full question & answer→Infinity or it does not oscillate.
Question 911 Mark
A driver wearing an electronic digital watch goes down into sea water with terminal velocity v. How will the time in the water proof watch be affected?
Answer
View full question & answer→ It will not be affected as its action is independent of gravity and buoyant force.
Question 921 Mark
What is the total energy of a simple harmonic oscillator?
Answer
View full question & answer→$\frac{1}{2}\text{m}\omega^2\text{r}^2$ where r = amplitude, $\omega$ = angular frequency, m = mass of the oscillator.
Question 931 Mark
A simple pendulum is mounted inside a space craft. What should be its time period of oscillation?
Answer
View full question & answer→ A simple pendulum is mounted inside a space craft. its time period of oscillation is given by$\text{T}=2\pi\sqrt{\frac{\text{l}}{\text{g}}}$
Here, I = 0, so T becomes infinity.
Here, I = 0, so T becomes infinity.
Question 941 Mark
Two springs of force constant $\mathrm{k}_1$ and $\mathrm{k}_2$ are joined in parallel. What is the force constant of the combination?
Answer
View full question & answer→Force constant $k$ of parallel combination is given by $k=k_1+k_2$. Thus, force constant of the parallel combination is equal to the sum of individual force constants of two springs.
Question 951 Mark
Are all periodic motions oscillatory by nature?
Answer
View full question & answer→All periodic motions need not be oscillatory.
Question 961 Mark
Will a pendulum gain or lose time when taken to the top of a mountain?
Answer
View full question & answer→At height as we move up 'g' decreases. Since $\text{T}\propto\frac{1}{\sqrt{\text{g}}}$ time period increases.
Question 971 Mark
How many times in one vibration, K.E. and P.E. become maximum?
Answer
View full question & answer→ Two times.
Question 981 Mark
A particle is executing $\text{S.H.M.}$ Identify the positions of the particle where:
- $\text{K.E}$. of the particle is zero.
- $\text{P.E}$. is zero.
- $\text{P.E}$. is one fourth of the total energy.
- $\text{P.E}$. and $\text{K.E}$. are equal.
Answer
View full question & answer→- At extreme position $(x = A)$
- At mean position $(x = 0)$
- $\frac{1}{2}\text{m}\omega^2(\text{A}^2-\text{x}^2)=\frac{1}{4}\times\frac{1}{2}\text{m}\omega^2\text{A}^2$
- $\frac{1}{2}\text{m}\omega^2(\text{A}^2-\text{x}^2)=\frac{1}{2}\text{m}\omega^2\text{A}^2$
Question 991 Mark
At what points is the energy entirely kinetic and potential in SHM?
Answer
View full question & answer→ At mean position, the energy is entirely K.E. At extreme positions, the energy is entirely P.E.
Question 1001 Mark
A mass of 2 kg is attached to the spring of spring constant $50 \mathrm{Nm}^{-1}$. The block is pulled to a distance of 5 cm from its equilibrium position at $\mathrm{x}=0$ on a horizontal frictionless surface from rest at $\mathrm{t}=0$. Write the expression for its displacement at anytime $t$.
View full question & answer→