# Linear Momentum of Unwinding Cylinder

• jamaicanking
In summary, the Linear momentum of an unwinding Cylinder with one end of a string attached to the ceiling can be found by using the equations for momentum and moment of inertia. The distance from the ceiling to the point on the cylinder is represented by l, and the linear momentum can be calculated by multiplying the mass of the cylinder by the derivative of the distance with respect to time. A free body diagram and Newton's Second Law can be applied to find the linear acceleration and ultimately determine the distance l(t).
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?

Last edited:
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).

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.

## 1. What is linear momentum of an unwinding cylinder?

The linear momentum of an unwinding cylinder refers to the tendency of the cylinder to continue moving in a straight line as it unwinds. It is a measure of the quantity of motion of the cylinder in a certain direction.

## 2. How is linear momentum calculated for an unwinding cylinder?

The linear momentum of an unwinding cylinder can be calculated by multiplying the mass of the cylinder by its velocity. This can be expressed mathematically as p = mv, where p is the linear momentum, m is the mass, and v is the velocity.

## 3. What factors can affect the linear momentum of an unwinding cylinder?

The linear momentum of an unwinding cylinder can be affected by factors such as the mass of the cylinder, its initial velocity, and any external forces acting on it. Friction between the cylinder and its surrounding surface can also affect its linear momentum.

## 4. Why is the linear momentum of an unwinding cylinder important?

The linear momentum of an unwinding cylinder is important because it helps to predict its future motion and behavior. It is also a fundamental principle in the study of mechanics and is used in various engineering and scientific applications.

## 5. How can the linear momentum of an unwinding cylinder be conserved?

The linear momentum of an unwinding cylinder can be conserved if there are no external forces acting on it. This means that the total linear momentum before and after the unwinding process should be equal. In real-world scenarios, external forces such as friction may cause some loss of linear momentum.

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