Question

Consider a mass-spring system. The spring has constant k=30N/m and the mass m=3kg. The mass oscillates with amplitude of 10cm.

What is the frequency of oscillation?

What is the displacement at time t=0?

When is the first time for the mass to be at maximum displacement? (t=?)

What is the maximum acceleration felt by the mass? Where in the motion does this occur?

What is the minimum acceleration felt by the mass? Where in the motion does this occur?

What is the maximum potention energy stored in the spring? (PE=1/2kx2)

Since energy is conserved, what is the maxium velocity of the mass? (KE=1/2mV2)

Answer 1

1)   

angular frequency w = sqrt ( k / m ) = sqrt ( 30 / 3 ) = 3.1623 rad / sec

frequency = w / (2pi) = 0.5033 Hz

2) At time t = 0 ... displacement = amplitude = 10 cm

3) the equation of motion is ...

x = 10 * cos ( 3.1623 * t )

first time = ... t = 0

4) maximum acceleration = w^2 * amplitude = 3.1623 ^2 * 0.1 = 1.000014129 m/sec2 ..

happens at t = 0 ... x = 10 cm

5) minimum acceleartion = 0 ...

happens at t = 0.496726 secs .... x = 0

6) maximum pot energy = 0.5 * k * amplitude ^2 = 0.5 * 30 * 0.1^2 = 0.15 J

7) 0.5 * m * v^2 = pot. energy of speing = 0.15

so max.. velocty = v = 0.31623 m/sec