Block A and B are connected with a cord. Block A has a mass of 20 kg. Block B ha

Question

Block A and B are connected with a cord. Block A has a mass of 20 kg. Block B has a mass of 10 kg. The blocks are initially at rest. Block A rests on a horizontal surface. The initial distance "a" between block A and the end of the spring is 1 meter. The coefficient of kinetic friction mu_k between block A and the surface is 0.1. The spring constant is 1000 N/m and the spring is initially uncompressed. The sequence of events are that block B moves down and block A moves to the right until it meets the spring. Block A will then compress the spring and B will continue to move down. Then the spring will extend and block A will move left and block B will move up. What is the velocity of block A just before it touches the spring? What is the maximum deflection of the spring? What is the velocity of block A as it just leaves contact with spring? What is the distance that A rebounds from spring before coming to rest?

Explanation / Answer

ma = 20 Kg
mb = 10 kg
a = 1 m
uk = 0.1
k = 1000 N/m

Mb*g - T = Mb*a
T - uk*Ma*g = Ma*a
T = Ma*a + uk*Ma*g  
Mb*g - (Ma*a + uk*Ma*g) = Mb*a
Mb*g - uk*Ma*g = (Ma + Mb)*a
a = (10*9.8 - 0.1*20*9.8)/(10 + 20)
a = 2.61 m/s^2

v = u + a*t
v = 0 + 2.61*1
v = 2.61 m/s
Velocity of block A just before touching Spring, v = 2.61 m/s

Now Using Energy Conservation,
Initial K.E of Block A & B + Initial P.E + Energy Lost in Friction = Final Spring Potential Energy + Final P.E of Block B
1/2*Ma*v^2 + 1/2*Mb*v^2 + P.Ein - P.Efin - uk*mA*g*x = 1/2*kx^2
1/2*Ma*v^2 + 1/2*Mb*v^2 + Mb*g*x - uk*mA*g*x = 1/2*kx^2
Substituing Values,
1/2*20*2.61^2 + 1/2*10*2.61^2 + 10*9.8*x - 0.1*20*9.8*x = 1/2*1000*x^2
x = 0.54 m
Maximum deflection of the spring, x = 0.54 m

Now, Again Using Energy Conservation,
Initial Spring Potential Energy + Initial P.E of Block B = Final K.E of A& B + Final P.E of Block B + Energy Lost in Friction
1/2*1000*0.54^2 = 1/2*20*v^2 + 1/2*10*v^2 + 10*9.8*0.54 + 0.1*20*9.8*0.54
v = 2.34 m/s

v^2 = u^ - 2*a*s
0 = 2.34^2 - 2*2.61*s
s = 1.05 m

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