Physics : Oscillations : Book Back Exercise, Example Numerical Question with Answers, Solution : Book Back Numerical Problems

__Oscillations (Physics)__

__Numerical Problems__

1. Consider the Earth as a homogeneous sphere of radius R and a straight hole is bored in it through its centre. Show that a particle dropped into the hole will execute a simple harmonic motion such that its time period is

Solution

Earth is assumed to be a homogeneous sphere.

Its centre is at O and Radius = R

The hole is bored straight through the centre along its diameter. The acceleration due to gravity at the surface of the earth = *g*

Mass of the body dropped inside the hole = *m*

After time t, the depth it reached (inside the earth) = *d*

The value of ‘g’ decreases with deportation.

So acceleration due to gravity at deportation = ‘*g*'

i.e.,g' = g(l -*d*//R) = g( (R-*d*) / R) ...(1)

Let *y* be the distance from the centre of the earth

Then y = Radius - distance = R - d

Substitute y in (1)

g' = g *y*/R

Now, force on the body of mass m due to this new acceleration g' will be

F = mg' = mg*y* /R

and this force is directed towards the mean position O.

The body dropped in the hole will execute S.H.M Spring factor k = *mg*/Radius

2. Calculate the time period of the oscillation of a particle of mass m moving in the potential defined as

where E is the total energy of the particle.

Solution

Length of simple pendulum *l* = 0.9 m

Inclined plane with the horizontal plane α = 45°

Time period of oscillation of simple pendulum T = ?

3. Consider a simple pendulum of length *l *= 0.9* m *which is properly placed on a* *trolley rolling down on a inclined plane which is at *θ* = 45° with the horizontal. Assuming that the inclined plane is frictionless, calculate the time period of oscillation of the simple pendulum.

Answer: 0.86 s

4. A piece of wood of mass m is floating erect in a liquid whose density is ρ. If it is slightly pressed down and released, then executes simple harmonic motion. Show that its time period of oscillation is

Solution:

Spring factor of liquid = Aρg

Inertra factor of wood piece = *m*

5. Consider two simple harmonic motion along *x* and *y*-axis having same frequencies but different amplitudes as *x *= A sin (ω*t *+ φ) (along* x *axis) and* y = *B sin ω*t* (along *y* axis). Then show that

and also discuss the special cases when

Note: *when a particle is subjected to two* *simple harmonic motion at right angle to each other the particle may move along different paths. Such paths are called Lissajous figures.*

Answer :

a. y=B/A x equation is a straight line passing through origin with positive slope.

b. y= - B/A x equation is a straight line passing through origin with negative slope.

c. equation is an ellipse whose center is origin.

d. *x*2+*y*2* *=* A*2, equation is a circle whose center is origin .

e. equation is an ellipse (oblique ellipse which means tilted ellipse)

6. Show that for a particle executing simple harmonic motion

a. the average value of kinetic energy is equal to the average value of potential energy.

b. average potential energy = average kinetic energy = ½ (total energy)

Hint : *average kinetic energy* = <*kinetic energy > = 1/T ∫0T*(*Kinetic energy*) *and average Potential energy = <Potential energy> =1/T ∫0T*(*Potential energy*)

7. Compute the time period for the following system if the block of mass m is slightly displaced vertically down from its equilibrium position and then released. Assume that the pulley is light and smooth, strings and springs are light.

Hint and answer:

Case(a)

Pulley is fixed rigidly here. When the mass displace by *y* and the spring will also stretch by *y*. Therefore, *F* = *T* = *ky*

Case(b)

Mass displace by *y*, pulley also displaces by *y. T* = 4*ky*.

Tags : Oscillations | Physics , 11th Physics : UNIT 10 : Oscillations

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11th Physics : UNIT 10 : Oscillations : Book Back Numerical Problems | Oscillations | Physics

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