An object of mass m = 3.0kg is released from rest at a height of h = 5.0m on an

Question

An object of mass m = 3.0kg is released from rest at a height of h = 5.0m on an inclined ramp with an angle of 10 degrees with the horizontal. At the foot of the ramp and after a horizontal surface of length d = 20m is the tip of a spring of force constant k = 4.0 x 103 N/m (see figure). The object slides down the ramp and into the spring, compressing it a distance x before coming momentarily to rest.

Find x and describe what happens to the object after it comes to a rest for the following cases:

a.) If all surfaces are frictionless

b.) If the coefficient of kinetic friction between the horizontal surface and the object is .20.

Thanks so much!

Explanation / Answer

a) if all surfaces are frictionless, then energy conserved. All the potential energy converted to the potential energy of the spring.

Therefore, mgh*Sin[] = (1/2)*kx^2 and x = (2mgh *Sin[]/k)

b) If the coefficient of kinetic friction between the horizontal surface and the object is .20, when the object stop, all potential energy converted to the the spring potential and the work done by the friction.

The friction is given by, Fs = N = *mg and the work done is W = Fd = *mg*d

Therefore, mgh*Sin[] = (1/2)*kx^2 + *mg*d

and we get, x = [2(mgh*Sin[] - *mg*d)/k]

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