The angular velocity of the spring be $\omega$ $x=a \cos (\omega t+\theta)$ At $t=0, x=x_{0}$ On Substituting these values in the above equation we get, $\mathrm{x}_{0}=\mathrm{A} \cos \theta-(1)$...

### A body describes simple harmonic motion with an amplitude of and a period of . Find the acceleration and velocity of the body when the displacement is 0

Amplitude is given as $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ Time period is given as $=0.2 \mathrm{~s}$ When the displacement is $y$, then acceleration is given as $A=-\omega^{2} y$ Velocity is given as...

### A body describes simple harmonic motion with an amplitude of and a period of . Find the acceleration and velocity of the body when the displacement is

(a)

(b)

Amplitude is given as $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ Time period is given as $=0.2 \mathrm{~s}$ When the displacement is $y$, then acceleration is given as $A=-\omega^{2} y$ Velocity is given as...

### A circular disc of mass is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be s. The radius of the disc is . Determine the torsional spring constant of the wire. (Torsional spring constant is defined by the relation , where is the restoring couple and the angle of twist).

Mass of the circular disc is given as $10 \mathrm{~kg}$ Period of torsional oscillation is given as $1.5 \mathrm{~s}$ Radius of the disc is given as $15 \mathrm{~cm}=0.15 \mathrm{~m}$ Restoring...

### Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

The mass of the particle executing simple harmonic motion is $m$. The particle's displacement at a given time $t$ is given by $x=A \sin \omega t$ Velocity of the particle is given as...

### You are riding in an automobile of mass . Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by during one complete oscillation. Estimate the values of (a) the spring constant and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports .

(a) Mass of the automobile is given as $=3000 \mathrm{~kg}$ The suspension sags by a length of $15 \mathrm{~cm}$ Decrease in amplitude $=50 \%$ during one complete oscillation If each spring's...

### An air chamber of volume V has a neck area of cross-section into which a ball of mass just fits and can move up and down without any friction. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Figure]

Solution: Volume of the air chamber is given as $\mathrm{V}$ Cross-sectional area of the neck is given as $\mathrm{A}$ Mass of the ball be $m$ The ball is fitted in the neck at position given as...

### One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Area of cross-section of the U-tube is given as $A$ Density of the mercury column is given as $\rho$ Acceleration due to gravity is given as $g$ Restoring force, F = Weight of the mercury column of...

### A cylindrical piece of cork of density of base area A and height h floats in a liquid of density . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period where is the density of cork. (Ignore damping due to viscosity of the liquid)

Base area of the cork is given as $=\mathrm{A}$ Height of the cork is given as $h$ Density of the liquid is given as $\rho_{1}$ Density of the cork is given as $\rho$ In equilibrium: Weight of the...

### A simple pendulum of length I and having a bob of mass is suspended in a car. The car is moving on a circular track of radius with a uniform speed If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

The centripetal acceleration supplied by the circular motion of the car, as well as the acceleration due to gravity, will be felt by the bob of the basic pendulum. Acceleration due to gravity is...

### Answer the following questions:

(a) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give the correct time during the free fall?

(b) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

(a) Wristwatches work on the principle of spring action and are not affected by gravity's acceleration. As a result, the time on the watch will be accurate. (b) The cabin's acceleration owing to...

### Answer the following questions:

(a) Time period of a particle in SHM depends on the force constant and mass of the particle: . A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small-angle oscillations. For larger angles of oscillation, a more involved analysis shows that is greater than Think of a qualitative argument to appreciate this result.

(a) The spring constant $k$ is proportional to the mass in the case of a simple pendulum. The numerator ($m$) and denominator ($d$) will cancel each other out. As a result, the simple pendulum's...

### The acceleration due to gravity on the surface of moon is . What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is ? (g on the surface of earth is )

Acceleration due to gravity on the surface of moon is given as $g^{\prime}=1.7 \mathrm{~m} \mathrm{~s}^{-2}$ Acceleration due to gravity on the surface of earth is given as $g=9.8 \mathrm{~m}...

### The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of . If the piston moves with simple harmonic motion with an angular frequency of , what is its maximum speed?

Angular frequency of the piston is given as $\omega=200 \mathrm{rad} / \mathrm{min}$ Stroke is given as$=1.0 \mathrm{~m}$ Amplitude is given as $A=1.0 / 2$ $=0.5 \mathrm{~m}$ The maximum speed...

### Figure (a) shows a spring of force constant k clamped rigidly at one end and a mass attached to its free end. A force applied at the free end stretches the spring. Figure (b) shows the same spring with both ends free and attached to a mass at either end. Each end of the spring in Fig. (b) is stretched by the same force F.

(a) What is the maximum extension of the spring in the two cases? (b) If the mass in Fig. (a) and the two masses in Fig. (b) is released, what is the period of oscillation in each case? Solution:...

### Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ( is in cm and is in .

(a) (b)

(a)$x=3 \sin (2 m t+\pi / 4)$ $=-3 \cos (2 \pi t+\pi / 4+\pi / 2)$ $=-3 \cos (2 \pi t+3 \pi / 4)$ $=-3 \cos (2 \pi t+3 \pi / 4)$ On comparing with the standard equation $A \cos (\omega t+\Phi)$, we...

### Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ( is in cm and is in .

(a)

(b)

(a) $x=-2 \sin (3 t+\pi / 3)$ $=2 \cos (3 t+\pi / 3+\pi / 2)$ $=2 \cos (3 t+5 \pi / 6)$ On comparing the above equation with the standard equation, $x=A \cos (\omega t+\Phi)$, Amplitude will be $A=2...

### The given figures correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the $x$-projection of the radius vector of the revolving particle $P$, in each case. Solution: (a) Time period is given as $t=2 \mathrm{~s}$...

### In the given figure, let us take the position of mass when the spring is unstreched as , and the direction from left to right as the positive direction of -axis. Give as a function of time for the oscillating mass if at the moment we start the stopwatch (t , the mass is at the maximum compressed position. In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Solution: The body is in the left position at maximal compression, with an initial phase of $3 \pi / 2$ rad. Then, $x=a \sin (\omega t+3 \pi / 2)$ $=-a \cos \omega t$ $=-2 \cos 20 t$ As a result,...

### In the given figure, let us take the position of mass when the spring is unstreched as , and the direction from left to right as the positive direction of -axis. Give as a function of time for the oscillating mass if at the moment we start the stopwatch (t , the mass is

(a) at the mean position,

(b) at the maximum stretched position.

Solution: Distance travelled by the mass sideways is given as $a=2.0 \mathrm{~cm}$ Angular frequency of oscillation can be calculated as, $\omega=\sqrt{k} / \mathrm{m}$ $=\sqrt{1200 / 3}$...

### A spring having with a spring constant is mounted on a horizontal table as shown in Fig. 14.24. A mass of is attached to the free end of the spring. The mass is then pulled sideways to a distance of and released.

Determine the maximum speed of the mass. Solution: Spring constant is given as $\mathrm{k}=1200 \mathrm{~N} \mathrm{~m}^{-1}$ Mass is given as $\mathrm{m}=3 \mathrm{~kg}$ Displacement is given as...

### A spring having with a spring constant is mounted on a horizontal table as shown in Fig. 14.24. A mass of is attached to the free end of the spring. The mass is then pulled sideways to a distance of and released.

Determine<br>(i) the frequency of oscillations, (ii) maximum acceleration of the mass Solution: Spring constant is given as $\mathrm{k}=1200 \mathrm{~N} \mathrm{~m}^{-1}$ Mass is given as...

### A spring balance has a scale that reads from 0 to . The length of the scale is A body suspended from this balance, when displaced and released, oscillates with a period of S. What is the weight of the body?

Maximum mass that the scale can rea is given as $M=50 \mathrm{~kg}$ Maximum displacement of the spring = Length of the scale, $I=20 \mathrm{~cm}$ $=0.2 \mathrm{~m}$ Time period is given as $T=0.6...

### The motion of a particle executing simple harmonic motion is described by the displacement function, If the initial (t ) position of the particle is and its initial velocity is , what are its amplitude and initial phase angle? The angular frequency of the particle is . If instead of the cosine function, we choose the sine function to describe the SHM: , what are the amplitude and initial phase of the particle with the above initial conditions. Solution:

At positlon, t = 0, The given function is $x(t)=A \cos (\omega t+\phi).....(1)$ $\begin{array}{l} 1=A \cos (\omega \times 0+\phi)=A \cos \phi \\ A \cos \phi=1 \end{array}$ Differentiating equation...

### Which of the following relationships between the acceleration a and the displacement of a particle involve simple harmonic motion?

(a)

(b)

The condition of SHM is stated below, Acceleration is directly proportional to negative of displacement of particle. If ' $\mathrm{a}$ ' is the acceleration, $\mathrm{x}$ is the displacement Then,...

### Which of the following relationships between the acceleration a and the displacement of a particle involve simple harmonic motion?

(a)

(b)

Condition of SHM is stated below, Acceleration is directly proportional to negative of displacement of particle. If ' $\mathrm{a}$ ' is the acceleration, $\mathrm{x}$ is the displacement Then, for...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at away from A going towards , and

(b) at away from B going towards .

(a) Positive, Positive, Positive From the end $A$, the particle is travelling toward point 0. This motion is going from $A$ to $B$, which is the standard positive direction. As a result, the...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the mid-point of AB going towards A,

(b) at away from B going towards

(a) Negative, Zero, Zero A basic harmonic motion is being performed by the particle. The particle's mean location is denoted by $O$. Its highest velocity is at the mean position $O$. Because the...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the end ,

(b) at the end B

(a) Zero, Positive, Positive Points A and B are the path's two ends, with A-B=10cm and'O' being the path's halfway. Between the end locations, a particle moves in a linear simple harmonic motion....

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $\exp \left(-\omega^{2} \mathrm{t}^{2}\right)$ It's a one-time exponential function that doesn't repeat. As a result, the motion is non-periodic. (b) The given function $1+\omega t+\omega^{2}...

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $3 \cos [4 \pi-2 \omega t]=3 \cos [2 \omega t-\pi / 4]$ The equation can be written in the form $\cos (\omega t+\phi)$. It is S.H.M with the period $2 \pi / 2 \omega=\pi / \omega$ (b) $\cos...

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $\sin \omega t-\cos \omega t$ =2[(1/2)sinωt-(1/2)cosωt] =\sqrt{2}[(1 / \sqrt{2}) \sin \omega t-(1 / \sqrt{2}) \cos \omega t] $=\sqrt{2}[\sin \omega t \times \cos (\pi / 4)-\cos...

### The figures depicts plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

(a) Because the motion is repeated in only one position, the depicted graph does not illustrate periodic motion. The full motion during one period must be repeated successively for a periodic...

### The figures depicts plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

(a) Motion is not periodic since it does not repeat itself after a set length of time. (b) The following graph depicts a periodic motion that repeats every 2 seconds.

### Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

(a) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.

(b) general vibrations of a polyatomic molecule about its equilibrium position.

(a) Simple harmonic motion (b) SHM is not periodic, although general vibrations of a polyatomic molecule about its equilibrium position are. The inherent frequencies of a polyatomic molecule are...

### Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

(a) the rotation of earth about its axis.

(b) motion of an oscillating mercury column in a U-tube.

(a) The earth's rotation is not a to-and-fro motion around a fixed point. As a result, it is regular but not S.H.M. (b) Simple harmonic motion

### Which of the following examples represents periodic motion?

(a) A hydrogen molecule rotating about its centre of mass.

(b) An arrow released from a bow.

(a) The rotation of a hydrogen molecule around its mass centre is periodic. This is due to the fact that when a hydrogen molecule spins about its centre of mass, it returns to the same location...

### Which of the following examples represents periodic motion?

(a) A swimmer completing one (return) trip from one bank of a river to the other and back.

(b) A freely suspended bar magnet displaced from its N-S direction and released.

(a) The motion of the swimmer is not regular. A swimmer's move between the sides of a river is to and fro. It does not, however, have a set duration. This is because the swimmer's back and forth...