# Fnet=ma 2008 #14: Kinetic Energy Increase with Angular Velocity

• dtl42
In summary, The spaceborne energy storage device initially has kinetic energy E and rotates at angular velocity ω. By reeling in the tether, the angular velocity increases to 2ω while the radius decreases by a factor of √2. This results in a new kinetic energy of 2E, since kinetic energy is proportional to the square of angular velocity and radius.
dtl42

## Homework Statement

A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy E and rotates at angular velocity ω. Energy is added until the device rotates at angular velocity 2ω. What is the new kinetic energy of the device?

Not sure.

## The Attempt at a Solution

I guessed that since kinetic energy is related to angular velocity squared, it would be 4E, but the answer is 2E.

v=rw and if the tether is reeled in r reduces as w increases.Angular momentum is conserved.

Thanks, I get it now.

Can somebody explain this in detail?

Well, as you're pulling in the tether, angular momentum is conserved. Angular momentum: $L=mvr$, so:

$$mvr_{1}=mvr_{2}$$
rewriting v as wr...
$$\omega_{1} r_{1}^{2}=\omega_{2} r_{2}^{2}$$

With $\omega_{2}=2\omega_{1}$, we get

$$r_{1}^{2}\omega_{1}=r_{2}^{2}\left(2 \omega_{1}\right)$$

$$\frac{r_{1}}{\sqrt{2}}=r_{2}$$

Kinetic energy...
$$K_{1}=mv^{2}=m\left(\omega r\right)^{2}=m\omega^{2}r^{2}$$

So, replacing with $\omega_{2}$ and $r_{2}$,

$$K_{2}=m\left(2\omega\right)^{2}\left(\frac{r}{\sqrt{2}}\right)^{2}=m\left(4\omega^{2}\right)\left(\frac{r^{2}}{2}\right)=2m\omega^{2}r^{2}=2K_{1}$$

## 1. What is Fnet=ma 2008 #14: Kinetic Energy Increase with Angular Velocity?

Fnet=ma 2008 #14 is a physics problem from the 2008 Advanced Placement Physics C exam. It involves calculating the increase in kinetic energy of a rotating object based on its angular velocity.

## 2. How do you calculate Fnet=ma 2008 #14?

To solve Fnet=ma 2008 #14, you will need to use the formula KE=1/2Iω^2, where KE is kinetic energy, I is the moment of inertia, and ω is angular velocity. You will also need to use the formula F=ma, where F is net force, m is mass, and a is acceleration. From there, you can use the given information in the problem to solve for the increase in kinetic energy.

## 3. What is angular velocity?

Angular velocity is a measure of how quickly an object is rotating or spinning. It is typically represented by the Greek letter ω and is measured in radians per second (rad/s).

## 4. How does angular velocity affect kinetic energy?

According to the formula KE=1/2Iω^2, the kinetic energy of a rotating object is directly proportional to the square of its angular velocity. This means that as angular velocity increases, so does the kinetic energy of the object.

## 5. What is the significance of Fnet=ma 2008 #14?

Fnet=ma 2008 #14 is a representative problem that tests your understanding of the relationship between angular velocity and kinetic energy. It also requires you to apply your knowledge of moment of inertia and net force to solve the problem. Understanding this relationship is crucial in understanding rotational motion and its effects on objects.

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