Unit 5 Overview: Rotation

This unit is allllll about objects that rotate! From a spinning record to a satellite, we can use this unit to describe the motion (or lack of) for these situations. Additionally, we will be making connections between this unit and previous units in ways such as demonstrating the analogs between translational and rotational kinematics.

Some Big Ideas outlined by College Board for this unit are:

1. • Force Interactions - Why does a curveball take less time to reach the plate than a fastball? Why is it easier to balance a bicycle when it's in motion?
2. • Conservation - How can you increase your swing on a swing set without being pushed?

Unit 5 will cover approximately 14%-20% of the exam and should take around 10 to 20, 45-minute class periods to cover. The AP Classroom personal progress check has 20 multiple choice questions and 1 free response question for you to practice on.

5.1 Torque and Rotational Statics

The first topic covered in Unit 5 is torque and rotational statics. Torque is defined as the force applied perpendicular to the radius of rotation, and it is measured in units of Newton-meters (N∙m). Rotational statics refers to the study of the equilibrium of rotating objects, which occurs when the sum of all torques acting on an object is zero. This topic includes the study of the center of mass and center of gravity, as well as the calculation of the torque required to maintain rotational equilibrium.

5.2 Rotational Kinematics

The next topic in Unit 5 is rotational kinematics, which is the study of rotational motion without considering the forces that cause it. This includes the study of angular displacement, angular velocity, and angular acceleration. These concepts are analogous to their linear counterparts, displacement, velocity, and acceleration, and are measured in units of radians, radians per second, and radians per second squared, respectively.

5.3 Rotational Dynamics and Energy

The third topic in Unit 5 is rotational dynamics and energy. This topic builds upon the concepts learned in the previous topic and introduces the study of the forces that cause rotational motion, including torque, moment of inertia, and rotational work and energy. This topic includes the study of rotational analogs of Newton’s laws, such as the angular version of Newton’s second law, and the rotational work-energy theorem.

5.4 Angular Momentum and Its Conservation

The final topic in Unit 5 is angular momentum and its conservation. Angular momentum is defined as the product of moment of inertia and angular velocity, and it is measured in units of kilogram-meters squared per second (kg∙m^2/s). This topic includes the study of the conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by an external torque. This principle has many important applications, including the explanation of the behavior of spinning objects, such as tops and gyroscopes.

Practice Problems

• A uniform rod of length L and mass M is pivoted at one end and allowed to swing freely in a vertical plane. What is the moment of inertia of the rod about the pivot point?
• A force of 10 N is applied tangentially to a wheel of radius 0.2 m. If the moment of inertia of the wheel is 0.1 kg·m², what is the resulting angular acceleration of the wheel?
• A merry-go-round of radius 3 m and mass 200 kg is rotating at 5 rad/s. What is the moment of inertia of the merry-go-round?
• A student pushes a 10-kg box along a horizontal floor with a force of 50 N. If the coefficient of static friction between the box and the floor is 0.4, what is the maximum torque the student can exert on the box before it begins to rotate?
• A flywheel of moment of inertia 2 kg·m² is rotating at 20 rad/s. How much work is required to bring the flywheel to a stop?

Answers:

1. The moment of inertia of a rod pivoted at one end is given by (1/3)ML². Therefore, the moment of inertia of the rod about the pivot point is (1/3)ML².
2. The torque applied to the wheel is equal to the force multiplied by the radius. Therefore, the torque is 2 N·m. The resulting angular acceleration is given by α = τ/I, where τ is the torque and I is the moment of inertia. Therefore, the angular acceleration is 20 rad/s².
3. The moment of inertia of a uniform disc is given by (1/2)MR², where M is the mass of the disc and R is the radius. Therefore, the moment of inertia of the merry-go-round is (1/2)(200 kg)(3 m)² = 900 kg·m².
4. The maximum torque that can be exerted on the box before it begins to rotate is equal to the product of the force of friction and the distance from the pivot point to the point of application of the force. Therefore, the maximum torque is 0.4(10 kg)(9.8 m/s²)(d), where d is the distance from the pivot point to the point of application of the force. Solving for d, we get d = 1.28 m. Therefore, the maximum torque is 62.72 N·m.
5. The work required to bring the flywheel to a stop is equal to the change in kinetic energy of the flywheel. Therefore, the work is (1/2)(2 kg·m²)(20 rad/s)² = 400 J.

Conclusion

In conclusion, Unit 5 of a typical physics curriculum is dedicated to the study of rotational motion. This unit covers topics such as torque and rotational statics, rotational kinematics, rotational dynamics and energy, and angular momentum and its conservation. These concepts are important for understanding the behavior of rotating objects, such as wheels, gears, and turbines, and have many important applications in engineering and science.