Linear Momentum of Unwinding Cylinder

[jamaicanking]

Homework Statement

Find the Linear momentum of an unwinding Cylinder with one end of a string attached to the ceiling. The mass of the cylinder is m and radius r. The cylinder is uniform and there is no slipping.

Homework Equations

Momentum, P = mv
moment of inertia , I = 1/2 MR^2

The Attempt at a Solution

distance from ceiling to point on cylinder , y = l(length of string unwound)

so is the momentum simply m*dl(t)/dt?

 

[kuruman]

Yes. Draw a free body diagram, apply Newton's Second Law for rotational and linear motion, find the linear acceleration and use it to find l(t).

 

[kerimek]

I would like to clarify that the linear momentum of the unwinding cylinder can be calculated using the formula P = mv, where m is the mass of the cylinder and v is the linear velocity of the cylinder. The linear velocity can be calculated using the relationship v = rω, where r is the radius of the cylinder and ω is the angular velocity. The angular velocity can be calculated using the formula ω = dθ/dt, where θ is the angular displacement and t is time. Since the cylinder is unwinding without slipping, the angular displacement can be related to the linear displacement using the formula θ = l/r, where l is the length of string unwound. Therefore, the linear momentum of the unwinding cylinder can be expressed as P = m(rω) = m(r dθ/dt) = m(r l/r)/dt = ml/t. This means that the linear momentum of the unwinding cylinder will increase as more string is unwound, and the rate of change of linear momentum will depend on the rate of unwinding (dl/dt) and the mass of the cylinder (m). Additionally, the moment of inertia of the cylinder can also be taken into account by using the formula P = Iω, where I is the moment of inertia and ω is the angular velocity. Overall, the linear momentum of the unwinding cylinder can be expressed as a function of time and can be calculated by considering the mass, radius, length of string unwound, and angular velocity of the cylinder.