Question

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.50 N is applied. A 0.500-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x 5 5.00 cm and released from rest at t 5 0. (a) What is the force constant of the spring? (b) What are the angular frequency v, the frequency, and the period of the motion? (c) What is the total energy of the system?

Answer 1

Answer:

a) 250 N/m

b) 22.4 rad/s , 3.6 Hz , 0.28 sec

c) 0.3125 J

Explanation:

a)

F = force applied on the spring = 7.50 N

x = stretch of the spring from relaxed length when force "F" is applied = 3 cm = 0.03 m

k = spring constant of the spring

Since the force applied causes the spring to stretch

F = k x

7.50 = k (0.03)

k = 250 N/m

b)

m = mass of the particle attached to the spring = 0.500 kg

Angular frequency of motion is given as

$w = \sqrt{\frac{k}{m}}$

$w = \sqrt{\frac{250}{0.5}}$

$w$ = 22.4 rad/s

$f$ = frequency

Angular frequency is also given as

$w$ = 2 π $f$

22.4 = 2 (3.14) f

$f$  = 3.6 Hz

$T$ = Time period

Time period is given as

$T = \frac{1}{f}$

$T = \frac{1}{3.6}$

$T$ = 0.28 sec

c)

A = amplitude of motion = 5 cm = 0.05 m

Total energy of the spring-block system is given as

U = (0.5) k A²

U = (0.5) (250) (0.05)²

U = 0.3125 J