A 1.80kg block slides with a speed of 0.960m/s on a frictionless horizontal surf

Question

A 1.80kg block slides with a speed of 0.960m/s on a frictionless horizontal surface until it encounters a spring with a force constant of 963N/m . The block comes to rest after compressing the spring 4.15 cm.

A) Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of 0 cm.

B) Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of 1.00 cm.

C) Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of 2.00 cm.

D) Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of 3.00 cm.

Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of 4.00 cm.

Explanation / Answer

Since there is no friction, total mechanical energy must be conserved. The block starts with energy of
K = (1/2)*m*v^2 = .9v^2

spring potential energy is given by
U = (1/2)*k*x^2 =481.5*x^2
where x is the instantaneous compression of the spring.

E= U+K

v^2-u^2=2ax
so at s=.0415
v=0
a=-.960^2/2/.0415
a=11.10m/s^2

so v=(.960^2-2*11.1*x)
   v=(.9216-22.2*x)^.5
so
A)
x=0m
v=.96m/s
U = 0J
K = .83J
E=.83J

B)
x=.01m
v=.84m/s
U=.048J
K=.635J
E=.683J
C)
x=.02m
v=.69m/s
U=.193J
K=.428J
E=.621J
D)
x=.03m
v=.51m/s
U=.433J
K=.234J
E=.667J

E)
x=.04m
v=.18m/s
U=.771J
K=.029J
E=.8J

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