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Q. 14.12 (d) Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t=0)$ 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: (x is in cm and t is in s).

(d)  $x=2\; cos\; \pi t$

The initial position of the particle is x(0) The radius of the circle i.e. the amplitude is 2 cm The angular speed of the rotating particle is  Initial phase is  The reference circle for the given simple Harmonic motion is

Q. 14.12 (c) Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $\inline (t=0)$ 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: (x is in cm and t is in s).

(c)  $\inline x=3\; sin(2\pi t+\pi /4)$

At t= 0    Reference circle is as follows

Q. 14.12 (b) Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t=0)$ 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: (x is in cm and t is in s).

(b)  $x=cos(\pi /6-t)$

The initial position of the particle is x(0) The radius of the circle i.e. the amplitude is 1 cm The angular speed of the rotating particle is  Initial phase is  The reference circle for the given simple Harmonic motion is

Q. 14.12 (a) Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t=0)$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: (x is in cm and t is in s).

(a)  $x=-2\; sin(3t+\pi /3)$

The initial position of the particle is x(0) The radius of the circle i.e. the amplitude is 2 cm The angular speed of the rotating particle is  Initial phase is  The reference circle for the given simple Harmonic motion is

Q. 14.11 Figures 14.25 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.

(a) Let the required function be   Amplitude = 3 cm = 0.03 m T = 2 s Since initial position x(t) = 0,  As the sense of revolution is clock wise Here x is in metres and t is in seconds. (b)Let the required function be   Amplitude = 2 m T = 4 s Since initial position x(t) = -A,  As the sense of revolution is anti-clock wise Here x is in metres and t is in seconds.

Q. 14.10 (c) In Exercise 14.9, let us take the position of mass when the spring is unstreched as $x=0,$and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch $(t=0),$ the mass is

(c) 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?

Amplitude is A = 0.02 m Time period is  (c) At t = 0 the mass is at the maximum compressed position. x(0) = -A Here x is in metres and t is in seconds. The above functions differ only in the initial phase and not in amplitude or frequency.

Q. 14.10 (b) In Exercise 14.9, let us take the position of mass when the spring is unstreched as $\inline x=0,$  and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch $\inline (t=0),$  the mass is

(b) at the maximum stretched position,

In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Amplitude is A = 0.02 m Time period is  (b) At t = 0 the mass is at the maximum stretched position. x(0) = A Here x is in metres and t is in seconds.

Q. 14.10 (a) In Exercise 14.9, let us take the position of mass when the spring is unstreched as $\inline x=0,$ and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch $\inline (t=0),$ the mass is

(a) at the mean position,

In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Amplitude is A = 0.02 m Time period is  (a) At t = 0 the mass is at mean position i.e. at t = 0, x = 0 Here x is in metres and t is in seconds.

Q. 14.9 (iii) A spring having with a spring constant $\inline 1200\; N\; m^{-1}$ is mounted on a horizontal table as shown in Fig. 14.24. A mass of $\inline 3\; kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of  $\inline 2.0 \; cm$  and released.

Determine

(iii) the maximum speed of the mass.

Maximum speed occurs at the mean position and is given by

Q. 14.9 (ii) A spring having with a spring constant $\inline 1200\; N\; m^{-1}$ is mounted on a horizontal table as shown in Fig. 14.24. A mass of $\inline 3\; kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $\inline 2.0\; cm$  and released.

Determine

(ii) maximum acceleration of the mass, and

A body executing S.H.M experiences maximum acceleration at the extreme points  (FA = Force experienced by body at displacement A from mean position)

Q. 14.9 (i) A spring having with a spring constant  $\inline 1200\; N\; m^{-1}$  is mounted on a horizontal table as shown in Fig. 14.24. A mass of $\inline 3\; kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $\inline 2.0\; cm$  and released.

Determine

(i) the frequency of oscillations,

The frequency of oscillation of an object of mass m attached to a spring of spring constant k is given by

Q. 14.8 A spring balance has a scale that reads from $0$ to $\inline 50\; kg.$ The length of the scale is $\inline 20 cm,$ A body suspended from this balance, when displaced and released, oscillates with a period of $\inline 0.6\; s.$ What is the weight of the body?

Spring constant of the spring is given by The time period of a spring attached to a body of mass m is given by

Q. 14.7  The motion of a particle executing simple harmonic motion is described by the displacement function,  $x(t)=A \; cos(\omega t+\phi ).$

If the initial $\inline (t=0)$  position of the particle is $\inline 1\; cm$ and its initial velocity is $\inline \omega \; cm/s,$  what are its amplitude and initial phase angle ? The angular frequency of the particle is  $\inline \pi s^{-1}.$ If instead of the cosine function, we choose the sine function to describe the SHM : $\inline x=B\; sin(\omega t+\alpha ),$  what are the amplitude and initial phase of the particle with the above initial conditions.

at t = 0 at t = 0 Squaring and adding equation (i) and (ii) we get Dividing equation (ii) by (i) we get at t = 0 at t = 0 Squaring and adding equation (iii) and (iv) we get Dividing equation (iii) by (iv) we get

Q. 14.6 Which of the following relationships between the acceleration a and the displacement $x$ of a particle involve simple harmonic motion?

(a)  $a=0.7x$

(b) $a=-200x^2$

(c) $a=-10x$

(d) $a=100x^3$

Only the relation given in (c) represents simple harmonic motion as the acceleration is proportional in magnitude to the displacement from the midpoint and its direction is opposite to that of the displacement from the mean position.

Q. 14.5 (f)  A particle is in linear simple harmonic motion between two points,$\inline A$ and $\inline B$, $\inline 10cm$ apart. Take the direction from $\inline A$ to$\inline B$ as the positive direction and  give the signs of velocity, acceleration and force on the particle when it is

(f) at $\inline 4cm$ away from $\inline B$ going towards$\inline A.$

Velocity, acceleration and force all are negative

Q.14.5 (e)  A particle is in linear simple harmonic motion between two points,$\inline A$ and $\inline B$, $\inline 10cm$ apart. Take the direction from $\inline A$ to $\inline B$ as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(e) at $\inline 3cm$ away from $\inline A$ going towards $\inline B$, and

Velocity is positive. Acceleration and force are also positive.

Q. 14.5 (d) A particle is in linear simple harmonic motion between two points, $\inline A$ and $\inline B$ , $\inline 10cm$ apart. Take the direction from $\inline A$ to $\inline B$ as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(d) at  $\inline 2cm$ away from$\inline B$ going towards $\inline A$,

Velocity is negative. Acceleration and force are also negative.

Q. 14.5 (c) A particle is in linear simple harmonic motion between two points,$\inline A$ and $\inline B$, $\inline 10 \; cm$ apart. Take the direction from $\inline A$  to $\inline B$ as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(c) at the mid-point of AB going towards $\inline A$,

Velocity is negative that is towards A and its magnitude is maximum. Acceleration and force are zero.

Q. 14.5 (b) A particle is in linear simple harmonic motion between two points, $\inline A$ and $\inline B$, $\inline 10cm$  apart. Take the direction from A to $\inline B$ as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(b) at the end $\inline B$,

Velocity is zero. Acceleration and force are negative.

Q. 14.5 (a) 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

(a) at the end A,

Velocity is zero. Force and acceleration are in the positive direction.
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