Earth spins around its own axis and orbits Sun. Calculate Earth's total kinetic energy and Earth's total angular momentum about Sun. Assume that Sun is so massive compared to Earth that the center of mass of the system is at Sun's center.

1a30001 Application of the angular momentum principle to a pendulum. 1a30001

A pendulum is composed of a 0.1-kg steel ball that is attached to the end of a 0.75 m long lightweight string. You pull the ball to the right to an angle of 30 and release it from rest.

What is the net torque on the ball about the pivot point at the instant it has been released?

What is the angular momentum of the ball relative to the pivot point 0.01 s after it was released?

What is the angular momentum of the ball about the pivot when it reaches its lowest point?

A rod can rotate about an axis perpendicular to the page and through the left end of the rod as shown. The forces shown in the image are applied to the rod. is applied at the end of the rod, is applied to the middle of the rod, and is applied at one-fourth the length of the rod. The length of the rod is 0.5 m. All forces have a magnitude of 75 N.

Figure: Forces applied to a rod.

What is the net torque on the rod?

1a10001 Angular momentum of a child running toward a merry-go-round 1a10001

A 20 kg child runs at a constant speed of 3.0 m/s, in a straight line tangentially to a merry-go-round, and jumps onto the edge of the merry-go-round of radius 2.0 m, as shown below.

Figure: A child running toward a merry-go-round.

What is the angular momentum of the child, relative to the center of the merry-go-round, just before landing on the merry-go-round?

What is her angular momentum, relative to the center of the merry-go-round, when she is 10.0 m from the edge of the merry-go-round?

What if she ran in the same direction with the same speed but jumped onto the "lower edge" of the merry-go-round in the picture, causing the merry-go-round to rotate counterclockwise?

1a70002 A baseball collides with a catching machine that rotates upon catching the ball. 1a70002

Here's a new invention for a "catching-machine'' that catches a baseball and tells you the speed of the ball based on how fast the machine rotates. The machine consists of very lightweight aluminum rods (they are so light that you can neglect their mass) connected to massive brackets with nets, 2 m from the center, that catch the baseball. The mass of each bracket at the end of a rod that holds a net is 10 kg. The entire machine sits on a low-friction axle fixed to the ground. When it catches a baseball, the machine's center of mass does not move. The machine's height is 1.0 m, and the ball travels nearly horizontally at the same height. The mass of the supporting rod is also negligible. See the pictures shown below.

Figure: Top view of the catching machine and baseball.

Figure: Side view of the catching machine and baseball.

If a baseball of mass 0.145 kg moving at 90 mph (40 m/s) in the direction shown in the figure below was caught by the machine, what would be the angular speed of the machine after catching the baseball?

If the collision takes place in 0.5 s, what is the net force on the system (of ball and machine) during the collision?

What is the net torque on the machine about its center as a result of the collision?

If the machine had "bouncy" pads mounted with springs (like little trampolines) instead of the nets, so that the baseball rebounds directly backward with the same speed, would the machine rotate faster or slower than when the ball is caught in a net? Explain your answer.

1a70001 A collision of a puck and a rotor. 1a70001

A puck moving on ice with a speed 0.5 m/s collides with a puck that is attached to another puck via a very lightweight rigid rod, as shown in the figure below. Let's refer to the attached pucks as a "rotor." After the collision, the incoming puck rebounds backward with a speed of 0.1 m/s and the rotor moves to the right and rotates clockwise. Kinetic energy is not conserved during the collision. All pucks have the same mass of 0.1 kg and the length of the rod is 0.8 m. Neglect the mass of the rod.

Figure: A puck collides with a rotor.

Using the momentum principle, calculate the velocity of the center of mass of the rotor after the collision. (Note: if the system is defined to be the set of three pucks, the net force on the system during the collision is zero.)

Using the angular momentum principle, what is the angular velocity vector for the rotor after the collision? (Note: if the system is defined to be the set of three pucks, the net torque on the system during the collision is zero.)

Using the energy principle, what is the change in the thermal energy of the system due to the collision? (Assume that Q=0 during the short time interval of the collision.)