Find the force P.

Question

A force P is applied to a string wrapped around the outer disk of a double pulley (mlarge = 5 kg, rlarge = 0.5 m, msmall = 2 kg, rsmall = 0.3

m). A string is wrapped around the inner disk with a 2 kg block attached to one end and a 4 N force applied to the other end. The strings do not slide on the pulleys. The axle supporting the pulley has a frictional torque of 5 Nm. The system is initially rotating in the clockwise direction at 3 rad/s, but is slowing down at a rate of 5 rad/s2. Find the force P.

Answer 1

If I understand the problem right, it is given to us that

M = 5 kg

R = 0.5 m

m = 2 kg

r = 0.3 m

m₀ = 2 kg

F = 4 N

τ₃ = 5 Nm

ω₀ = 3 rad/s

α = 5 rad/s²

P - ?

To solve the problem, we will use equation

(τ_M + τ_P + τ₃ + τ_F) - (t _m + τ_m0) = αI

Torque of force P is τ_P = PR,

Torque of force F is τ_F = FR

Torque of outer disk τ_M = MgR

Torque of inner disk t _m = mgr

Torque of block τ_m0 = m₀gr

Torque of friction τ₃ = 5 Nm

Total moment of inertia

I = I₁ + I₂

Moment of inertia of big disk I₁ = 1/2 MR²,

Moment of inertia of small disk I₂ = 1/2 mr²

So,

I = 1/2 (MR² + mr²)

Solving all equations for the force P, we finally will have

P = ((1/2(α (MR² + mr²)) + gr(m + m₀) - R(Mg + F) - τ₃ )) / R

I would like you to put in the values of variables into the equation above and get the numeric answer.