Question

a. Traveling up the incline, the work done on the block by friction is

b. What is the block’s velocity when it has traveled a distance D=1 meter up the incline?

c. If mass m₁ is cut in half, speed of the block when it has traveled D=1 meter up the incline
   a.stays the same
   b. increases
   c. decreases

Answer 1

a) Traveling up the incline, the work done on the block by friction is

W = -μ_kmgDcosθ

where μ_k = Coefficient of friction, D = Distance travelled, θ = Angle of inclined plane

b) As the block moves up the inclined plane, there are two forces that cause it to decelerate. These forces are the component of its weight that is parallel to the inclined plane and the friction force.

Force parallel = mg*sinθ
Ff = μ_k*mg*cosθ
Total force = mg*sinθ + μ_k*mg*cosθ

To determine the rate at which the velocity is decreasing, divide this force by the block’s mass.

a = g*sinθ + μ_k*g*cosθ  

To determine the block’s velocity after moving 1 meter up the incline, use the following equation.

vf^2 = vi^2 + 2 * a * d
vf^2 = vi^2 + 2*-(g*sinθ + μ_k*g*cosθ)*1
vf^2 = vi^2 + 2*- g*sinθ - μ_k*g*cosθ
vf = √(vi^2 - 2g*sinθ - 2μ_k*g*cosθ)

c) As velocity is independent of mass so it will remain same.