A 10.0 kg mass is suspended on an incline by a rope connected to a wall

Question

A 10.0 kg mass is suspended on an incline by a rope connected to a wall. The coefficients of static and kinetic friction between the mass and the incline are 0.65 and 0.50, respectively. 

a) Draw a free-body diagram of the mass.

b) Determine the tension of the rope.

c) If the rope is cut, determine if the mass makes it to the bottom of the incline and if so how long it takes to reach the bottom. Assume the mass is 1.20 m from the bottom of the incline.

   

Answer 1

On free body diagram you must show 4 forces:

1. mg - weight; directed vertically down,
2. N - normal reaction force; directed up perpendicular to the surface,
3. T - tension; directed towards the rope, parallel to the incline up,
4. f_s - static frictional force; directed parallel to the incline up.

Because the mass is not moving, we can write

mg + N + T + f_s = 0

Let's chose the coordinate system so, that x-axis is parallel to the incline up, y- axis is perpendicular to it and has the same direction as normal force N. If we know angle Θ between incline and horizontal, then we can write in scalar form

x: T + f_s - mg·sin Θ = 0 (1)

y: N - mg·cos Θ = 0 (2)

Because static frictional force

f_s = μ_s·N,

using equation (2), we get

f_s = μ_smg·cos Θ

If we put it in (1) and solve against tension, we will have

T = mg(sin Θ - μ_s·cos Θ) (3)

From formula (3) we see that we need to know angle Θ to find numeric value of tension T.

To answer (c) we need to compare angle of incline Θ with the maximum angle that the incline could have

with horizontal before the body starts to slide. We can find this angle using equation

tan Θ_max = μ_s (4)

It is easy to find out that if the angle of incline with horizontal greater than 33°, the body will start to move down incline if the rope is cut.

Then using the formula

t = √ 2d /a (5)

we will find the time of sliding.

In the formula d = 1.20 m and

a = g(sin Θ - μ_k·cos Θ) (6)

Please pay attention to the fact that in formula (6) μ_k is coefficient of kinetic friction.