Problems on Applications of SHM
Q1

A block of unknown mass is attached to a spring of spring constant 6.5 N/m and undergoes simple harmonic motion with an amplitude of 10cm. When the mass is halfway between its equilibrium position and the end point, its speed is 30 cm/s. Find the mass of the block and the time period of the motion ?

Sol:

The spring constant of the spring k = 6.5 N/m
Amplitude of the simple harmonic motion A = 10 cm = 0.1 m
Velocity at x = = 0.05m is v = 30 cm/s = 0.3 m/s
But velocity at any point in the SHM is given by

V =
ω2 =
At x = , v = 0.3 m/s
ω2 =
= 12
Q2

A mass M is attached to a spring. It oscillates every 2 sec. If the mass is increased by 5 kg, the period increases by 3 sec. Find the initial mass M, assuming that Hooke's law is obeyed.

Sol:

Time period of oscillation T =
Given Time period, T = 2 sec (when m = M)
Time period, T = 2 + 3 sec (when m = M + 5) = 5 sec
Dividing these two equations, we get

Q3

A mass m is oscillating freely on a vertical spring, when m = 0.81 kg , the period is 0.91 sec. An unknown mass on the spring has a period of 1.16 sec. Find the spring constant and the unknown mass?

Sol:

mass m = 0.81 kg
Time period T = 0.91 sec
Time period for unknown mass, m is T = 1.16 sec

Q4

A body of mass 12 kg is suspended by a coil spring of natural length 50 cm and force constant 2 × 103 N/m. What is the stretched length of the spring ? If the body is pulled down further stretching the spring to 59 cm and then released, what is the frequency of oscillations of the suspended mass ?

Sol:

Mass of the body m = 12 kg
Length of the spring L = 50 cm = 0.5 m
Force (constant of the spring k) = 2 × 103 N/m
Let l be the increase in length of the spring.
Then the restoring force balances the weight (as the system is in equilibrium).
Stretched length of the spring = L + l = 0.558m
frequency of oscillation f,
The frequency of oscillation is independent of stretch in the spring

Q5

Two springs have their spring constants k1 and k 2 (k1 > k2). Determine the spring on which more work is done when:
(i) Their lengths are increased by the same amount
(ii) They are stretched by the same force

Sol:

(i) Let each spring be stretched through a distance x.
work done on first spring =
work done on second spring =
As k1 > k2, >
Therefore more work is done on the first spring

(ii) Let each spring is stretched by the same force, and x1 and x2 be the extension in each spring then

x1 =
x2 =
work done on first spring =
=
=
work done on second spring =
=
=

As k1 > k2, <
Work done on the second spring is more if they are stretched by same force

Q6

A block of mass m is placed on a frictionless surface and is connected to two springs of force constants k1 and k2. calculate the time period of the oscillation of the block?

Sol:

Suppose the block is displaced to the right through a small distance x.
Then the right spring gets compressed and the left one elongates.
Let F1 and F2 be the restoring forces in the springs of force constants k1 and k2 respectively.
Then F1 = −k1 x and F2 = − k2 x.
Total restoring force F = F1 + F2
= −(k1 + k2) x
Where k = k1 + k2 is the force constant of the whole system.
Time period of the oscillation, T