Question

An oscillating block-spring system has a mechanical energyof 1.00 J and amplitude of 10.0 cm and a maximum speed of 1.20m/s. Find (a) the spring constant and (b) the mass of theblock, and (c) the frequency of oscillation.

Answer 1

Step-by-step explanation:

Given

Mechanical Energy of Spring-Block system is 1 J

Maximum Amplitude is
`A=10 cm`

maximum speed
`v_(max)=1.2 m/s`

Suppose
`x=A\sin \omega t`be general equation of motion of spring-mass system

where A=max amplitude

`\omega`=Natural frequency of oscillation

t=time

`v_(max)=A\omega =1.2`

`0.1\cdot \omega =1.2`

`\omega =12 rad/s`

maximum kinetic Energy must be equal to total Mechanical Energy when spring is un deformed i.e. at starting Position

`(1)/(2)mv_(max)^2=1`

`m=(2)/(1.2^2)=(2)/(1.44)=1.38 kg`

Also
`\omega`is also given by

`\omega =\sqrt{(k)/(m)}`

`k=\omega ^2\cdot m`

where k= spring constant

`k=12^2\cdot 1.38`

`k=200 N/m`