Exploration 1: Constant Angular Velocity Equation
By now you have seen the equation: \(\theta =\theta_{0}+ \omega_{0}\ast t\). Perhaps you have even derived it for yourself. But what does it really mean for the motion of objects? This Exploration allows you to explore both terms in the equation: the initial angular position by changing \(\theta_{0}\) from \(0\) radians to \(6.28\) radians and the angular velocity term by changing \(\omega_{0}\) from \(-15\text{ rad/s}\) to \(15\text{ rad/s}\). Restart.
Answer the following questions (position is given in meters and time is given in seconds).
- How does changing the initial angular position affect the motion?
- How does changing the initial angular velocity affect the motion?
Exploration 2: Constant Angular Acceleration Equation
By now you have seen the equation: \(\theta =\theta_{0}+\omega_{0}\ast t + 0.5\ast\alpha\ast t^{2}\). Perhaps you have even derived it for yourself. But what does it really mean for the motion of objects? This Exploration allows you to explore all three terms in the equation: the initial angular position by changing \(\theta_{0}\) from \(0\) radians to \(6.28\) radians, the angular velocity term by changing \(\omega_{0}\) from \(-15\text{ rad/s}\) to \(15\text{ rad/s}\), and the angular acceleration by changing \(\alpha\) from \(-5\text{ rad/s}^{2}\) to \(5\text{ rad/s}^{2}\). Restart.
Answer the following questions (position is given in meters and time is given in seconds).
- How does changing the initial angular position affect the motion of the object?
- How does changing the initial angular velocity affect the motion of the object?
- How does changing the angular acceleration affect the motion of the object?
- Can you get the object to change direction?
Exploration 3: Torque and Moment of Inertia
A mass (between \(0.01\text{ kg}\) and \(1\text{ kg}\)) is hung by a string from the edge of a massive (between \(0\text{ kg}\) and \(2\text{ kg}\)) disk-shaped pulley (with a radius between \(0.1\) and \(4\) meters) as shown (position is given in meters, time is given in seconds, and angular velocity is given in radians/second). Restart.
Set the hanging mass to \(0.25\text{ kg}\), the radius of the pulley to \(2\text{ m}\), and vary the mass of the pulley.
- How does the magnitude of the angular acceleration of the pulley depend on the mass (and therefore moment of inertia) of the pulley?
- How does the magnitude of the acceleration of the hanging mass depend on the mass (and therefore moment of inertia) of the pulley?
- How are your answers to (a) and (b) related?
Set the mass of the pulley to \(0.5\text{ kg}\), the radius of the pulley to \(2\text{ m}\), and vary the hanging mass.
- How does the magnitude of the angular acceleration of the pulley depend on the hanging mass?
- How does the magnitude of the acceleration of the hanging mass depend on the hanging mass?
- How are your answers to (d) and (e) related?
Set the hanging mass to \(0.25\text{ kg}\), the mass of the pulley to \(0.5\text{ kg}\), and vary the radius of the pulley.
- How does the magnitude of the angular acceleration of the pulley depend on the radius of the pulley?
- How does the magnitude of acceleration of the hanging mass depend on the radius of the pulley?
- How are your answers to (g) and (h) related?
Set the mass of the pulley to \(0.5\text{ kg}\), the hanging mass to \(0.25\text{ kg}\), and the radius of the pulley to \(2\text{ m}\).
- Determine the acceleration of the hanging mass and the angular acceleration of the pulley.
- From Newton's second law, determine the tension in the string.
- How much torque does this tension provide to the pulley?
Exploration by Chuck Niederriter and Mario Belloni.
Exploration 4: Torque on Pulley Due to the Tension of Two Strings
Shown is a top view of a pulley on a table. The massive disk-shaped pulley can rotate about a fixed axle located at the origin. The pulley is subjected to two forces in the plane of the table, the tension in each rope (each between \(0\text{ N}\) and \(10\text{ N}\)), that can create a net torque and cause it to rotate as shown (position is given in meters, time is given in seconds, and angular velocity is given in radians/second). Restart. Also shown is the "extended" free-body diagram for the pulley. In this diagram the forces in the plane of the table are drawn where they act, including the force of the axle.
Set the mass of the pulley to \(1\text{ kg}\), the radius of the pulley to \(2\text{ m}\), vary the forces, and look at the "extended" free-body diagram.
- How is the force of the axle related to the force applied by the two tensions?
- How do you know that this must be the case?
Set the mass of the pulley to \(1\text{ kg}\), the radius of the pulley to \(2\text{ m}\), and vary the forces.
- What is the relationship between \(F_{1}\) and \(F_{2}\) that ensures that the pulley will not rotate?
- For \(F_{1} > F_{2}\), does the pulley rotate? In what direction?
- For \(F_{1} < F_{2}\), does the pulley rotate? In what direction?
- What is the general form for the net torque on the pulley in terms of \(F_{1}\), \(F_{2}\), and \(r_{\text{pulley}}\)?
Set the mass of the pulley to \(1\text{ kg}\), \(F_{1}\) to \(10\text{ N}\), \(F_{2}\) to \(5\text{ N}\), and vary the radius of the pulley.
- How does the angular acceleration of the pulley depend on the radius of the pulley?
Set the radius of the pulley to \(2\text{ m}\), \(F_{1}\) to \(10\text{ N}\), \(F_{2}\) to \(5\text{ N}\), and vary the mass of the pulley.
- How does the angular acceleration of the pulley depend on the mass of the pulley?
- Given that the pulley is a disk, find the general expression for the angular acceleration in terms of \(F_{1}\), \(F_{2}\), \(m_{\text{pulley}}\), and \(r_{\text{pulley}}\).
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.
