# Velocity of center of mass of spinning disk

• atlarge
In summary, a solid uniform disk of mass 19.0 kg and radius 70.0 cm is at rest on a frictionless surface. A string is wrapped around the rim of the disk and a constant force of 35.0 N is applied to the string. When the disk has moved a distance of 6 m, the speed of the center of mass is found to be 5.1 m/s. This is calculated using Newton's Second Law, taking into account that only the force applied by the string contributes to the acceleration of the disk.

#### atlarge

A solid uniform disk of mass 19.0 kg and radius 70.0 cm (.7 m) is at rest FLAT on a frictionless surface. A string is wrapped around the rim of the disk and a constant force of 35.0 N is applied to the string. The string does not slip on the rim. The string is being pulled. When the disk has moved a distance of 6 m, what is the speed of the center of mass?

Relevant equations (I think...):
I=1/2MR^2
FR=Iα
α=a/R

So far, I have:
FR=Iα
FR=1/2MR^2(a/R)
F=1/2Ma
a=2*F/M=3.33333 m/s^2

and then,
v^2=2*a*6m, since initial velocity is 0.
v=7.26 m/s

But this is wrong, and I'm not sure why. Please help!

Newton's second law holds for translational motion, even if the net force results in a torque. The angular and linear components can be dealt with separately when they are not "coupled" via some friction or other mechanical mechanism (like a rolling object or a pulley and string).

Thank you. I'm not sure how I would use Newton's Second Law, due to the fact that the normal force and mg are perpendicular to the movement and therefore not contributing to the velocity.

I'm so frustrated :)

atlarge said:
Thank you. I'm not sure how I would use Newton's Second Law, due to the fact that the normal force and mg are perpendicular to the movement and therefore not contributing to the velocity.

I'm so frustrated :)

mg and the normal force are equal and opposite, so their contribution vanishes. You only have to deal with the force applied via the string.

OH, thank you gneill!

F=ma
35N=(21kg)a
a=1.667 m/s^2

v^2=0 + 2(1.667)(7.9), 0 being the initial velocity
v=5.1 m/s

## 1. What is the concept of center of mass in a spinning disk?

The center of mass of a spinning disk is the point at which the disk's mass can be considered to be concentrated. It is the point around which the disk's mass is evenly distributed, and it remains fixed even as the disk spins.

## 2. How is the velocity of the center of mass of a spinning disk calculated?

The velocity of the center of mass of a spinning disk can be calculated using the formula v = ωr, where v is the velocity, ω is the angular velocity, and r is the distance from the center of mass to the axis of rotation.

## 3. How does the velocity of the center of mass change as the spinning disk rotates?

The velocity of the center of mass remains constant as the spinning disk rotates, as long as there are no external forces acting on the disk. This is due to the conservation of angular momentum.

## 4. What factors can affect the velocity of the center of mass of a spinning disk?

The velocity of the center of mass of a spinning disk can be affected by the mass distribution of the disk, the angular velocity of the disk, and any external forces acting on the disk, such as friction or air resistance.

## 5. How does the velocity of the center of mass of a spinning disk relate to its rotational kinetic energy?

The velocity of the center of mass of a spinning disk is directly proportional to its rotational kinetic energy. This means that as the velocity increases, so does the rotational kinetic energy, and vice versa.