**Question** 14.4:

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 (ω is any positive constant):

(a) sin *ωt *– cos *ωt*

(b) sin^{3} *ωt*

(c) 3 cos (π/4 – 2*ωt*)

(d) cos ω*t *+ cos 3*ωt *+ cos 5*ωt*

(e) exp (–*ω*^{2}*t*^{2})

(f) 1 + *ωt *+ *ω*^{2}*t*^{2}

**Answer**:

**(a) **SHM

The given function is:

This function represents SHM as it can be written in the form:

Its period is:

**(b) **Periodic, but not SHM

The given function is:

sin3ωt=143sinωt-sin3ωt

The terms sin *ωt *and sin *ωt* individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

**(c) **SHM

The given function is:

This function represents simple harmonic motion because it can be written in the form:

Its period is:

**(d) **Periodic, but not SHM

The given function is . Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.

**(e) **Non-periodic motion

The given function is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

**(f) **The given function 1 + *ωt* + *ω*^{2}*t*^{2} is non-periodic.

**Question** 14.5:

A particle is in linear simple harmonic motion between two points, A and B, 10 cm 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,

(b) at the end B,

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

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

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

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

**Answer**:

**Answer:**

**(a) **Zero, Positive, Positive

**(b) **Zero, Negative, Negative

**(c) **Negative, Zero, Zero

**(d) **Negative, Negative, Negative

**(e) **Positive, Positive, Positive

**(f) **Negative, Negative, Negative

**Explanation:**

The given situation is shown in the following figure. Points A and B are the two end points, with AB = 10 cm. O is the midpoint of the path.

A particle is in linear simple harmonic motion between the end points

**(a) **At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point.

Its acceleration is positive as it is directed along AO.

Force is also positive in this case as the particle is directed rightward.

**(b)** At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point.

Its acceleration is negative as it is directed along O.

Force is also negative in this case as the particle is directed leftward.

**(c)**

The particle is executing a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is negative as the particle is directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position.

**(d)**

The particle is moving toward point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to B. Hence, the particle’s velocity and acceleration, and the force on it are all negative.

**(e)**

The particle is moving toward point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the values for velocity, acceleration, and force are all positive.

**(f)**

This case is similar to the one given in** (d)**.

**Question** 14.6:

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

(a) *a *= 0.7*x*

(b) *a *= –200*x*^{2}

(c) *a *= –10*x*

(d) *a *= 100*x*^{3}

**Answer**:

(c) A motion represents simple harmonic motion if it is governed by the force law:

*F* = –*kx*

*ma* = –*k*

Where,

*F* is the force

*m* is the mass (a constant for a body)

*x* is the displacement

*a* is the acceleration

*k* is a constant

Among the given equations, only equation *a* = –10 *x *is written in the above form with Hence, this relation represents SHM.

**Question** 14.7:

The motion of a particle executing simple harmonic motion is described by the displacement function,

*x* (*t*) = *A *cos (ω*t *+ φ).

If the initial (*t *= 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s^{–1}. If instead of the cosine function, we choose the sine function to describe the SHM: *x = B sin *(*ωt *+ α), what are the amplitude and initial phase of the particle with the above initial conditions.

**Answer**:

__Initially, at t = 0:__

Displacement, *x* = 1 cm

Initial velocity, *v* = ω cm/sec.

Angular frequency, ω = π rad/*s*^{–1}

It is given that:

Squaring and adding equations (*i*) and (*ii*), we get:

Dividing equation (*ii*) by equation (*i*), we get:

SHM is given as:

Putting the given values in this equation, we get:

Velocity,

Substituting the given values, we get:

Squaring and adding equations (*iii*) and (*iv*), we get:

Dividing equation (*iii*) by equation (*iv*), we get:

**Question** 14.8:

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

**Answer**:

Maximum mass that the scale can read, *M* = 50 kg

Maximum displacement of the spring = Length of the scale, *l* = 20 cm = 0.2 m

Time period, *T* = 0.6 s

Maximum force exerted on the spring, *F* = *Mg*

Where,

*g* = acceleration due to gravity = 9.8 m/s^{2}

*F *= 50 × 9.8 = 490

∴Spring constant,

Mass *m*, is suspended from the balance.

Time period,

∴Weight of the body = *mg *= 22.36 × 9.8 = 219.167 N

Hence, the weight of the body is about 219 N.

**Question** 14.9:

A spring having with a spring constant 1200 N m^{–1} is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

**Answer**:

Spring constant, *k* = 1200 N m^{–1}

Mass, *m *= 3 kg

Displacement, *A* = 2.0 cm = 0.02 cm

**(i)** Frequency of oscillation *v*, is given by the relation:

Where, *T* is the time period

Hence, the frequency of oscillations is 3.18 cycles per second.

**(ii)** Maximum acceleration (*a*) is given by the relation:

*a* = ω^{2} *A*

Where,

ω = Angular frequency =

*A* = Maximum displacement

Hence, the maximum acceleration of the mass is 8.0 m/s^{2}.

**(iii)** Maximum velocity, *v*_{max} = *A*ω

Hence, the maximum velocity of the mass is 0.4 m/s.

**Question** 14.10:

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

(a) at the mean position,

(b) at the maximum stretched position, and

(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?

**Answer**:

**(a) ***x = *2sin 20*t*

**(b) ***x* = 2cos 20*t*

**(c) ***x* = –2cos 20*t*

The functions have the same frequency and amplitude, but different initial phases.

Distance travelled by the mass sideways, *A* = 2.0 cm

Force constant of the spring, *k* = 1200 N m^{–1}

Mass, *m* = 3 kg

Angular frequency of oscillation:

= 20 rad s^{–1}

**(a)** When the mass is at the mean position, initial phase is 0.

Displacement,* x* = *A*sin *ωt*

*= *2sin 20*t*

**(b)** At the maximum stretched position, the mass is toward the extreme right. Hence, the initial phase is .

Displacement,

= 2cos 20*t*

**(c)** At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is .

Displacement,

= –2cos 20*t*

The functions have the same frequency and amplitude (2 cm), but different initial phases .