# CLEAR Calculus

## Lab 3: The physics of rotation

“δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω”
“Give me a place to stand, and I shall move the world.”
- Archimedes

1. The angular velocity $$\pmb{\omega}$$ of rotation of a rigid body has direction equal to the axis of rotation and magnitude equal to the rate of spinning measured in radians per second. The sense of $$\pmb{\omega}$$ is determined by the right-hand rule: if your right hand fingers curl in the direction of rotation, your thumb gives the direction of $$\pmb{\omega}$$ as shown in the figure below.

Let $$\bf r$$ be a vector from a point $$O$$ on the axis to a point $$P$$ on the rigid body.
1. What is the radius of the circular path traveled by $$P$$?
2. Show that the quantity $$\bf v=\pmb{\omega}\times\bf r$$ is the velocity of $$P$$.
Hint: You need to check both the length and the direction.

2. A door knob is located as far as possible from the hinge line for a good reason. If you want to open a heavy door, Where you apply a force and in what direction you push are important. Consider a vector force $$\bf F$$ acting on a body that is free to rotate about a point $$O$$. The force is applied at a point $$P$$ whose position determines the vector $$\bf r$$. The directions of $$\bf F$$ and $$\bf r$$ make an angle of $$\varphi$$ with each other as in the figure below.
1. Decompose the force into a radial component $${\bf F}_r$$ (in the direction of $$\bf r$$ ) and the tangential component $${\bf F}_t$$ (in the direction of motion). That is, write the vectors  $${\bf F}_r$$ and  $${\bf F}_t$$ in terms of the other vectors and quantities provided. Which of these components acts to move the body? What happens to the other component of $$\bf F$$?
2. Define torque as the vector $$\pmb{\tau}= \bf r\times \bf F$$. Derive a formula for $$||\pmb{\tau}||$$ in terms of $$||\bf F_t||$$ and $$||\bf r||$$.
3. What happens to $$||\pmb{\tau}||$$ if $$||\bf r||$$ is doubled? What does this correspond to with a door?
4. What happens to $$||\pmb{\tau}||$$ if $$||\bf F_t||$$ is doubled? What does this correspond to with a door?
5. What happens to $$||\pmb{\tau}||$$ if $$||\bf F_r||$$ is doubled? What does this correspond to with a door?
6. Sometimes torque is described as "force times lever arm." What does this mean?