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

  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.
    angular velocity 
    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?