A 3-kg block is launched from rest at point A by a spring of spring constant k1

Question

A 3-kg block is launched from rest at point A by a spring of spring constant k1 = 64 N/m onto a (mostly) frictionless track. The only section of track with friction is right after point B. the block slides along the track, up over a hill (point C) and then collides with another spring at point D, causing the block to turn around and go the other way.

a. The block is found to be traveling at a speed of 4 m/s when it gets to point C (at the top of the hill). If the first spring is initially compressed by an amount of delta s = 50 cm, find the height H of the block at point A.

b. Find the normal foce (magnnitude and direction) exerted by the track on the block at point C.

c. The spring at point D has spring constant k2 = 42 N/m. How far will the spring be compressed before the block comes to a momentary stop?

Explanation / Answer

(a) Using consevation of energy,

Total energy at C and A (energy is equal)

1/2m*v^2 + m*g*h -Fr*deltaX = m*g*H+1/2*k*s^2

1/2v^2 + g*h -mu*g*deltaX = g*H+1/2*k*s^2/m

1/2*4^2 +9.81*2-0.3*9.81*1.5=9.81*H+1/2*64*0.5^2/3

8+19.62-4.41=9.81*H+2.67

H = 2.09 m

(b) normal force,

N = m*g = 3*9.81 =29.43 N

c. 1/2*k'*s'^2 = 3*(8+19.62-4.41)

s'^2 = 2*3*(8+19.62-4.41)/42

=3.32

s' =1.82 m

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