### Calculus 3 Labs

## 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

- 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.

- What is the radius of the circular path traveled by \(P\)?
- 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.

- What is the radius of the circular path traveled by \(P\)?
- 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.
- 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 \(\bf F\)? - 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 \(||\bf r||\).
- What happens to \(||\pmb{\tau}||\) if \(||\bf r||\) is doubled? What does this correspond to with a door?
- What happens to \(||\pmb{\tau}||\) if \(||\bf F_t||\) is doubled? What does this correspond to with a door?
- What happens to \(||\pmb{\tau}||\) if \(||\bf F_r||\) is doubled? What does this correspond to with a door?
- Sometimes torque is described as "force times lever arm." What does this mean?

- Decompose the force into a