Question

The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz.

Answer 1

Answer:

(a) 9.375 N/m

(b) 0.5024 m/s

(c) 0.01 kg/s

Explanation:

mass of head, m = 15 g = 0.015 kg

frequency, f = 4 Hz

Time period, T = 1 / f = 0.25 s

Let k is the spring constant.

(a)

The formula for the time period is

$T=2\pi\sqrt{\frac{m}{K}}$

$0.25=2\times 3.14 \sqrt{\frac{0.015}{K}}$

$0.04=\sqrt{\frac{0.015}{K}}$

K = 9.375 N/m

(b)

Amplitude, A = 2 cm

Let ω is the angular velocity.

Maximum velocity, v = A ω = A x 2πf

v = 0.02 x 2 x 3.14 x 4 = 0.5024 m/s

(c)

Let b is the damping constant.

A(t = 4s) = 0.5 cm

Ao = 2 cm

Using the formula of damping

$\frac{A}{A_{0}}=e^{-\frac{bt}{2m}}$

$\frac{0.5}{2}}=e^{-\frac{b\times 4}{2\times 0.015}}$

$0.25=e^{-133.3 b}$

Taking natural log on both the sides

ln (0.25) = - 133.3 b

- 133.3 b = - 1.386

b = 0.01 kg/s