A block of mass m rests on a plane inclined at an angle with the horizontal. A s

Question

A block of mass m rests on a plane inclined at an angle with the horizontal. A spring with force constant k is attached to the block. The coefficient of static friction between the block and plane is s. The spring is pulled upward along the plane very slowly.

(a) What is the extension of the spring the instant the block begins to move. (Use any variable or symbol stated above along with the following as necessary: g.)
d =  

(b) The block stops moving just as the extension of the contracting spring reaches zero. Express k (the coefficient of kinetic friction) in terms of s and .
k =  

Explanation / Answer

The relevant equations

F = -K*x

Esys = Emech + Etherm + Echem + Eother

U = U0 + mgh

Spring potential energy U = (1/2)K*x^2

Kinetic Energy K = (1/2)m*v^2

Fn = m*g*cos(theta)

(a) X forces

-K*x, Fn*us and m*g*sin(theta)
  
y forces

Fn and Fg*cos(theta)

sum Fx = -Fn*us - m*g*sin(theta) + (-K*x)

solve for x

x = (-m*g/K)*[us*cos(theta)+sin(theta)]

(b) E1 = (1/2)m*v^2 + (1/2)K*x^2

E2 = m*g*h + fk*x

fk*x = Fn*uk*x

= m*g*uk*x*cos(theta)

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