Optional Practical: Conservation of angular momentum (Algodoo)
Angular velocity is defined as the product of the moment of inertia and the angular velocity. If no external torques act then the angular momentum of a body will remain constant, this means that if its moment of inertia decreases its angular velocity will increase. This is why the angular velocity of ice skater (Yuna Kim) in the video clip increases when she brings her leg and body closer to her axis of rotation.
The following instructions are intended to be used by students who are already familiar with the basic features of Algodoo.
Mass on a spring
In this exercise a mass will be rotated on the end of a stiff spring. The reason a spring will be used is because its length can be changed so we can easily vary its rotational inertia. You might wonder why we don't simply vary the mass but if you try this artificial example you will find that it doesn't work, the angular velocity is not affected.
- Open a new scene in Algodoo delete the ground then turn off gravity and air resistance.
- Using the circle tool place a circle of about 0.2 m radius onto the workspace.
- Add a horizontal spring joining the circle to the background.
- Right click the circle and set its mass to 1 kg
- drag the velocities window to one side so it stays visible.
- Set the vertical component of the velocity to 1 ms-1 .
- Right click the spring and drag its properties window to one side, set the length to 1m and damping and spring constant to maximum
- Run the simulation.
When a point mass, m moves in a circle radius, r with speed, v the angular momentum = mvr
- Try changing the length of the spring and observe what happens to the velocity.
- Use the conservation of angular momentum to find the speed when the length is 2m, verify your answer with the simulation.
- Predict the speed at a length of 0.1m. Does your simulation give the correct result, if not why do you think this is?
- Why were damping and spring constant set to maximum values?
- By displaying the information about the circle, monitor the KE as the length is varied. Where does the energy go to and come from?
In the next exercise the angular momentum of two coaxial disks will be monitored as they rotate relative to each other.
- Draw a circle of radius 1.5 m, in "appearance" tick "circle cake" this makes it easier to see the centre.
- Set the mass of the circle to 2 kg.
- Calculate the moment of inertia of the disk (I = 1/2Mr2)
- Draw a second circle of radius 1 m and give it a circle cake.
- Set the mass of the second disk to 1 kg and calculate its moment of inertia
- Place the small disk on top of the large one so the centres are coincident.
- Join the disks by putting an axle at their centre.
- Double click the axle and tick motor then set the motor speed to 5 rpm.
- Run the animation.
- The motor turns the turns the small wheel but why does the big one rotate in the opposite direction?
- Display the velocities of the two circles and calculate the angular momentum of the disks.
- Check your answer by looking at the "information" of the circles.
- Try adjusting the motor speed observing what happens to the angular momentum of the disks.
The ball and rod
Consider the example of a ball hitting a rod like in the example shown in the animation. At the start there is no rotation but at the end there is. How can angular momentum be conserved?
In examples like this we must look at the motion of the ball about the centre of the rod L = mvr where r is the perpendicular distance between the direction of motion of the ball and the centre of the rod. Now you will recreate this simulation and see how angular momentum is conserved.
- Draw a vertical rod 4 m long and set its mass to 1.5 kg and restitution to 0.09
- Calculate the moment of inertia of the rod about its centre.
- Switch on the grid and tick "snap to".
- Draw a small circle 1 m above the centre of the rod.
- Set the mass of the circle to 0.1 kg and its restitution to 0.03
The reason the restitution is set so low is to stop the ball bouncing back off the rod, this makes the calculation a bit easier. Actually if you watch the animation you will see that the ball appears to keep moving after the collision however it is not moving relative to the centre of the rod. You can view from the rods frame of reference by right clicking the rod and ticking "follow" in the "selection" options.
- Calculate the angular momentum of the ball.
- Display the velocities of the rod.
- Run the animation and calculate the final angular momentum of the rod from its angular velocity.
- Is momentum conserved?
- Try for other examples.