Question

1. An object of mass 10 kg is released at point A, slides to the bottom of the 30° incline, then collides with a horizontal massless spring, compressing it a maximum distance of 0.75 m. The spring constant is 500 N/m, the height of the incline is 2.0 m, and the horizontal surface is frictionless. (a) What is the speed of the object at the bottom of the incline? (b) What is the work of friction on the object while it is on the incline? (c) The spring recoils and sends the 2.0 m object back toward the incline. What is the speed of the object when it 30° MW reaches the base of the incline? (d) What vertical distance does it move back up the incline?

Answer 1

(a). Let v be the velocity of the object at the bottom of the incline. The kinetic energy at that time will be (1/2) m v². When the object reaches the spring and pushes it, this kinetic energy is transformed to the potential energy of the spring.
The potential energy of the spring, U = (1/2) k x²
Where x is the compression and k is the spring constant.
Thus we can equate the kinetic energy of the object and the potential energy of the spring to find the velocity
(1/2) m v² = (1/2) k x²
v = x sqrt (k/m)
v = 0.75 m x sqrt (500 N/m / 10 kg )
v = 5.30 m/s
(b) Using the conservation of energies we can find the work done by the friction.
The final kinetic energy when the object reaches the bottom is equal to the difference between the initial potential energy when the object was at the top of the incline and the work done by the friction
KE = U_i - W_f
(1/2) m v² = m gh - W_f ---------------------------- 1
(1/2) x 10 kg x (5.30 m/s)² = 10 kg x 9.8 m/s² x 2.0 m - W_f
W_f = 55.37 J
This is the work done by the friction.
(c) Since there is no friction along the horizontal surface, the object feels no force along the motion. Thus the object will hit the base of the incline with the same velocity as it started.
v = 5.30 m/s
(d) Since the incline offers the same frictional force and the equation 1 can be used again to find the maximum height reached, we will get the same height as every quantity is the same. Thus the maximum height it reaches is the maximum height of the inciline.
h = 2 m