In the past year or two,
radio-controlled cars and airplanes have been joined by a new competitive
sport: battling robots. These gadgets are designed to tear one another limb
from limb... or rather gear from gear. Trying to keep a robot functioning for
three whole minutes in the ring can be quite a challenge when the opponent's
robot is going after it with pick axes, spinning saw blades, or even flipping
wedges. Doing lots of damage to the other robot is largely a matter of
controlling three important quantities: energy, momentum, and angular momentum.
You've
watched enough robot battles to have an idea of what works and what doesn't.
Although the robots are all battery powered, their batteries can't supply enough
energy in a short period of time to do much damage to an opponent. A
well-designed robot accumulates energy from its batteries in something else
that can transfer a huge amount of energy in an instant. You're up to the
challenge, so you decide to build a robot that swings a massive spiked ball on
the outer edge of a horizontal metal disk.
1. Your robot's spiked ball swings around in a horizontal circle as an electric motor spins the disk. When electric power is supplied to the motor, it exerts a constant torque on the disk & ball. Assuming that there is no friction or air resistance, (a) how does the disk & ball's angular acceleration change with time and (b) how does the disk & ball's angular velocity change with time?
2. When the disk is spinning fast enough, you turn off the electric motor. The disk and ball are now free of torques. (a) How does the disk & ball's angular momentum change with time and (b) how does the disk & ball's angular velocity change with time?
3. The ball is attached to the outer edge of the disk by a short chain. With the motor still turned off, you drive your robot close to your opponent and press the "extend chain" button on your radio controller. The chain suddenly gets longer, so that the ball is now farther from the center of the spinning disk. (a) How does the disk & ball's angular momentum change with time and (b) how does the disk & ball's angular velocity change with time?
4. When the chain gets long enough, the ball collides with your opponent's robot and knocks it completely out of the ring. (a) Use the concept of impulse to show that the impact of the moving ball with the initially stationary opponent's robot transfers momentum to that robot (be sure to include the direction of that transferred momentum) and (b) use the concept of work to show that the impact of the moving ball with the initially stationary opponent's robot transfers energy to that robot.
Your robot won the tournament
but you are disconsolate at all the destruction it wrought on so many cute and
cuddly robots. You retire its spike ball to the trophy case and decide to apply
your robot and your intellect to more peaceful and compassionate pursuits. Your
robot begins its new existence as a messenger system in your local hospital. It
delivers flowers and medicines to children on a pediatric ward and is the
highlight of the day for many of the patients.
5. When you drive your robot slowly and carefully, its wheels don't skid on the floor. Why does that mean that it doesn't waste any of its energy producing heat with its wheels?
6. On the other hand, if you try to accelerate forward too quickly, the wheels skid. Why does this skidding (a) waste energy and (b) reduce the rate at which the robot accelerates forward, as compared to the fastest it could accelerate forward if its wheels hadn't started skidding?
7. Your robot stops to pick up a bunch of daisies and then heads off to a patient's room. As it starts heading toward that room, (a) what does the robot do to its wheels that causes the robot to be pushed forward and (b) what pushes the robot forward?
8. After dropping off the flowers, your robot drives out into the sloping hallway. You turn off the wheel motors and let it roll down the ramp-like hallway on its own, to the delight of the children watching. Use the concepts of potential energy and acceleration to explain why the robot naturally accelerates down the ramp rather than up it.