### Center of Mass of the Map of Texas

A card-board cutout of Texas has various holes around its perimeter. It is loosely hung from a rod. When the map of Texas is in equilibrium, it means that the pivot point is directly above the center of mass (CM). So, we use a string and a bob to draw a line on the map straight down, meaning we must have drawn over the CM. If we do this two times, the lines will intersect at the CM. This result can be verified by tossing and spinning the Map of Texas, as it will rotate about its center of mass.

### Center of Mass Toys

In this video, we showcase various toys that demonstrate the concept of center of mass. These items will be shown in stable equilibrium. Any small disturbance will result in a restoring torque which will bring the body back to the original equilibrium point.

### Skyhook

A skyhook is impossible to balance on the tip of your finger because of the torque created by force of gravity. When the belt is hanging over the skyhook, the center of mass of the system is shifted under the pivot point (the tip of the finger) due to the weight of the buckle. As a result, the system remains in stable equilibrium.

### Grip Bar

The grip bar has hooks placed along the bar at different radius from the pivot point, or handle. When the weight is hooked on just below the pivot, it is easier to hold the bar perfectly flat because although the weight is heavy it is contributing an almost zero torque to the bar with respect to the pivot point. As the weight is placed along the bar further away from the pivot point, it produces a greater torque and it becomes more and more difficult to hold the bar horizontally.

### Sledge Hammer

A plank of wood has a hinge screwed into it which is screwed into another plank of wood. One of the planks is fastened to a table while the other plank is off of the table. The other plank is held by hand and released, causing it to fall down. A rope is attached to the end of the wooden plank. When the sledge hammer is placed in the rope and left to dangle underneath the wood, the heaviest part of the system becomes the sledge hammer's head. The center of mass of the system is shifted to the left of the pivot point (the hinge). Therefore, the torque due to gravity is directed counterclockwise about the pivot point and the system remains in equilibrium.

### Angular Momentum: Rotating Stool

This demonstration is an example of conservation of angular momentum. We first start off with a rotating stool and some heavy weights. Once the presenter spins himself, with the weights close to his chest, he will start off with some initial angular velocity. As he extends his arms out, his moment of inertia is increased, and in order to conserve angular momentum, his angular velocity decreases. When the arms are brought back in, the moment of inertia decreases and angular velocity increases. In the second experiment, we demonstrate that angular momentum is a vector quantity, so that means that it has a direction. The presenter grabs a bike wheel and spins it, giving it some angular momentum. The presenter flips the spinning bike wheel upside down, changing the direction of its angular momentum. To conserve the angular momentum of the entire system, the presenter spins in the opposite direction of the bike wheel.

### Rotational Inertia

This experiment is a simple demonstration of rotational inertia and its relationship to angular acceleration and torque. We will be using the same mass throughout the experiment and hanging it off a string and pulley system to apply torque to the system. We first start off with the masses at the far ends of the arms, thus having a large moment of inertia. The pulley-mass system then applies some torque to the apparatus, causing it to spin at some rate. If we move the masses on the arm closer to the center, and apply the same torque, the apparatus will spin faster due to its smaller moment of inertia.

### Downhill Race With Cookie Cans

One metal cookie can has magnets placed along the rim, whereas the second can has them placed in the middle. They start rolling downhill from the top of the inclined plane. Assuming rolling without slipping, their potential energy due to gravity gets converted into the kinetic energy of the rotational motion and the kinetic energy of center-of-mass motion. The cookie can with weights along its rim has a higher moment of inertia; therefore, it will have a higher rotational kinetic energy and lower kinetic energy of center-of-mass motion. Therefore, the cookie can with weights located in the center will have a higher center of mass velocity and will win the race.

### Bike Wheel

When the bike wheel does not rotate, it simply falls down due to the torque created by the force of gravity. Once the bike wheel is set spinning, it has a large angular momentum along its axis of rotation. In this case the torque due to gravity causes the angular momentum and therefore the wheel axis to rotate (precess) around the vertical direction.

### Gyroscope

When the gyroscope does not rotate it behaves as one might expect under the force of gravity: it simply falls down. When the Gyroscope axis is set spinning, it will have a large angular momentum along the rotation axis. As the weights are added to one end of the axis, they create a torque due to gravity with respect to the pivot point, which causes the angular momentum and therefore the gyroscope axis to rotate (precess) around the vertical direction.

### Static Equilibrium

We will use this demonstration to learn about static equilibrium. An object in a state of static equilibrium is stationary and all forces and torques acting on it are balanced. Our apparatus consists of a torque beam, a sliding mass, a string and force sensor. One end of the torque beam is attached to a pivot and the other end is held up by the string. The tension in the string is measured by the force sensor. The sliding mass is exerting a torque on the beam, while the tension in the string is counter-balancing this torque to keep the system in a state of static equilibrium. The further the mass is away from the pivot, the more torque it applies, and the more torque the tension has to provide in order to counterbalance. This can be seen in the graph of the tension force vs. position of the sliding mass.

### Torque Window Washer

A beam with a sliding mass is supported by two strings on each end. Their tension is measured by red and blue force sensors. The graph on the top left indicates these tension forces (with colors matching the sensors) depending on the position of the sliding mass. The system is in static equilibrium, meaning that all forces and torques are balanced. When the sliding mass is slid towards one end, the equilibrium condition gives that the closer string should have a larger tension.