Q1. A windmill spins with a linear velocity of 6.3 m/s and has a radius of 25 m.

• a) How many rotations occur per minute?

• b) What is the angular acceleration if in 10 s the linear velocity changes to 2.6 m/s?

• c) With the acceleration found in part b: what is the angular velocity after 123 s?

Background

Topic: Rotational Kinematics

This question tests your understanding of the relationship between linear and angular velocity, angular acceleration, and rotational motion calculations.

Key Terms and Formulas:

• Linear velocity (): How fast a point on the edge of the windmill moves.

• Angular velocity ():

• Angular acceleration ():

• Rotations per minute (rpm):

• Angular velocity after time:

Step-by-Step Guidance

1. For part a: Calculate the initial angular velocity using with m/s and m.

2. Convert angular velocity from radians per second to rotations per minute using .

3. For part b: Find the final angular velocity using m/s and m.

4. Calculate the average angular acceleration using , where and s.

5. For part c: Set up the equation for angular velocity after 123 s using with the values found above.

Try solving on your own before revealing the answer!

Q2. A bowling ball is rolled towards the pins at 17 mph with 202.143 Joules of energy and a diameter of 20 cm.

• a) What is the mass of the ball?

• b) What is the new kinetic energy if the mass were doubled?

Background

Topic: Rotational and Translational Kinetic Energy

This question tests your ability to calculate kinetic energy for a rolling object, considering both translational and rotational components.

Key Terms and Formulas:

• Kinetic energy ():

• Moment of inertia for a solid sphere:

• Relationship for rolling without slipping:

• Combined kinetic energy:

Step-by-Step Guidance

1. Convert the velocity from mph to m/s if needed for SI units.

2. Use the combined kinetic energy formula for a rolling sphere: .

3. Set up the equation with the given kinetic energy and solve for mass .

4. For part b: Double the mass and recalculate the kinetic energy using the same formula.

Try solving on your own before revealing the answer!

Q3. A physics student lies on a lightweight plank supported by two scales 2.50 m apart. The left scale reads 300 N (head side), and the right scale reads 122 N.

• a) Find the student's mass.

• b) Find the distance from the student's head to her center of mass.

• c) If force 1 is removed and the board is allowed to fall, what is the angular acceleration if the student is treated as a long thin rod?

Background

Topic: Center of Mass, Torque, and Rotational Dynamics

This question tests your understanding of equilibrium, center of mass, torque, and rotational acceleration.

Key Terms and Formulas:

• Sum of forces:

• Torque equilibrium:

• Moment of inertia for a rod:

• Angular acceleration:

Step-by-Step Guidance

1. For part a: Add the forces from both scales to find the total weight, then use to solve for mass.

2. For part b: Set up the torque equilibrium equation to solve for the center of mass location relative to the head.

3. For part c: Use the moment of inertia formula for a rod and calculate the torque about the pivot point, then set up the angular acceleration equation.

Try solving on your own before revealing the answer!

Q4. Your ship, The Flamingo, has a mass of kg and is attempting to split two asteroids with masses kg and . M1 and M2 are separated by 16 miles, and The Flamingo is 15 miles from the centerline.

• a) What are the component forces of gravity experienced by The Flamingo?

Background

Topic: Newton's Law of Universal Gravitation and Vector Components

This question tests your ability to calculate gravitational forces and resolve them into components.

Key Terms and Formulas:

• Gravitational force:

• Component forces: ,

• Angle calculation:

Step-by-Step Guidance

1. Calculate the distance from The Flamingo to each asteroid using the Pythagorean theorem.

2. Find the angle for the force components using .

3. Calculate the gravitational force from each asteroid using .

4. Resolve each force into and components using and .

5. Add the and components from both asteroids to find the net force vector.

Try solving on your own before revealing the answer!