Nicholas H. answered • 07/04/22
Medical student with background in physics and chemistry
The key points to this problem are the work-energy theorem (the change in kinetic energy is equal to the net work done by all forces acting on the object) and the relationship between conservative forces and potential energy (the negative of the change in potential energy is equal to the work done by conservative forces).
For the first part, the forces acting on the block are the spring force, gravity, and a normal force. Since all forces doing work on the block are conservative (meaning that they conserve energy), we can use the conservation of mechanical energy. The block starts at rest and ends at rest, so the change in kinetic energy is 0, thus we only need to consider the potential energy.
The gravitational kinetic energy is given by U = mgh. Since the block is restricted to moving along the inclined plane, the change in height, h, is related to the distance traveled by the block, d, by h = d*sin(θ). The elastic potential energy is given by U = (1/2)kx^2. The displacement of the spring is equal to the distance the block travels down the incline. We can say the block at the top has only gravitational potential energy and the block at the bottom of the incline has only elastic potential energy. Using this, we can equate the final potential energy to the initial potential energy:
mgh = (1/2)kx^2 , replacing h and x to get terms of d
mgd*sin(θ) = (1/2)kd^2, dividing both sides by d
mg*sin(θ) = (1/2)kd, with m = 3, g = 9.8 or 10, θ = 30, and k = 80 this can be solved for d.
For the second part, the only non conservative force doing work is friction. We can start with ΔK = net work done = work done by conservative forces + work done by non conservative forces = -ΔU + work done by friction. Again, the block begins and ends at rest, so ΔK = 0, so work done by friction (W) = ΔU.
The work done by friction is equal to the force of friction times the distance the block travels in the direction of that force (since friction will act in the opposite direction as the movement, the work will be negative). W = -µmg*cos(θ)d.
Combining out equations and using our potential energies from the first part:
-µmg*cos(θ)d = (1/2)kd^2 - mgd*sin(θ), dividing both sides by d
-µmg*cos(θ) = (1/2)kd - mg*sin(θ), rearranging
mg(sin(θ) - µ*cos(θ)) = (1/2)kd, dividing both sides by (1/2)k
(mg(sin(θ) - µ*cos(θ))) / ((1/2)k) = d
Notice that this second answer is the same as the first if µ = 0 and will decrease as µ increases which should fit with your intuition regarding friction. Further, we can find a restriction on µ where d = 0 which corresponds to the value of µ where the gravitational force is not strong enough to break the friction holding the block in place.
