1. A 200 g mass resting on a frictionless surface is attached to a spring with a

Question

1. A 200 g mass resting on a frictionless surface is attached to a spring with a stiffness of 40.0 N/m. At t = 0 it is 0.100 m to the left of the equilibrium position and traveling to the right at 1.58 m/s.

(a) What are the amplitude and frequency of the oscillation of the mass? (b) Find the phase constant for the oscillation.

(c) Write the position as a function of time, x(t), for the oscillation.


Now the mass in question is passing through the equilibrium position going to the right it collides with a 50.0 g dart going to the left at 10 m/s. The dart embeds itself in the mass.

(d) Find the velocity of the mass and dart just after the dart hits it.

(e) Find the new amplitude and frequency of the motion of the mass and dart.

Explanation / Answer

a) use energy
PE=PE+KE
(1/2)kx2=(1/2)kx2+(1/2)mv2

20x2=20(.1)2+.1(1.58)2

x2=.022482

x=.150m

b)

x=.15sin((k/m)t+)

we put in what we know when t=0

.1=.15sin((200)*0+)

sin=(2/3)

=.72973 radians

c)

x(t)=.15sin((200)t+.72973)

d)use conservation of momentum

the velocity of the cart at the middle is the product of the amplitude and angular frequnecy

vc=2.1213 m/s

now apply conservation of momentum:

.2(2.1213)-.05(10)=v(.25)

v=-.3029 m/s (negative means to the left)

e)now we do energies

PE=KE

(1/2)kx2=(1/2)mv2

20x2=.125(.3029)2

x=.02395m

Hope that helps

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