N"Simple Pulley Problem: Find Tension in String for 100N Toy on Rough Surface"

[Lokhtar]

Homework Statement

A child has a toy of weight 100N which is attached to a string. The toy is initially at rest on a rough horizontal plane. The child picks up the string and pulls it so that the string makes an angle of 16 degrees with vertical. If the coefficient of friction between the toy and the floor is 0.8 (μ) and the toy is in limitng equilibrium, find the tension in the string.

The Attempt at a Solution

Hv=(Hcos16)-100+r=0
R=(Hcos16)-100

Hh=(Hsin16)-f
Hsin- μr=0
(Hsin16)-0.8r=0
(Hsin16)-0.8[(Hcos16)-100]=0
(Hsin16)-(0.8Hcos16)+80=0
H(sin16-0.8cos16)=-80
H=-80/sin16-(0.8cos16)
H=134.37

 

[ehild]

The Attempt at a Solution

R=(Hcos16)-100

The sign of R is wrong.

ehild

 

[kerimek]

N

I would first like to clarify that the given problem is not a simple pulley problem, as it involves a rough surface and a coefficient of friction. The solution provided in the attempt is also incorrect, as the equation for horizontal equilibrium is missing the tension (T) in the string.

To solve this problem, we can use the concept of Newton's laws of motion. The toy is in limiting equilibrium, which means that the forces acting on it are balanced. Therefore, we can set up the following equations:

Vertical equilibrium: Tcos16 - 100 = 0
Horizontal equilibrium: Tsin16 - μN = 0

Where N is the normal force exerted by the surface on the toy and μ is the coefficient of friction. We can solve these equations simultaneously to find the tension in the string (T).

Substituting the value of N from the second equation into the first equation, we get:

Tcos16 - 100 = Tsin16 - μ(Tcos16 - 100)

Solving for T, we get T = 134.37N.

Therefore, the tension in the string is 134.37N. It is important to note that this is the minimum tension required to keep the toy in limiting equilibrium. If the tension is lower than this value, the toy will start to slide on the rough surface.