Unit 10: Fixed-Axis Rotation
Angular kinematics, moment of inertia built by integration, torque, and rolling motion as translation and rotation combined.
9 topics · 32 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Angular Position, Velocity, and Acceleration
- How it worksHow do angular position, angular velocity, and angular acceleration mirror their linear counterparts exactly?
- Worked exampleFind the angular velocity of a wheel that completes 5 revolutions in 2 seconds.
- Common mistakeWhy isn't linear speed the same for every point on a spinning rigid body, even though they all share one angular velocity?
Your turn1 of 2
A wheel completes 9 revolutions in 4 s. What is its angular velocity, in radians per second?Relating Linear and Angular Variables
- How it worksHow do v = r omega and a = r alpha connect a point's linear motion to the rotation of the whole rigid body?
- Worked exampleFind the linear speed of a point 0.3 m from the axis of a wheel spinning at 20 rad/s.
- Going furtherWhat happens to a point's linear speed if it sits exactly on the rotation axis of a spinning object?
Your turn1 of 2
A point is 0.587 m from the axis of a wheel spinning at 28 rad/s. What is its linear speed, in meters per second?Rotational Kinematics With Constant Angular Acceleration
- How it worksHow do the constant-angular-acceleration equations mirror the constant-linear-acceleration equations rung for rung?
- Worked exampleFind the angle a wheel turns through while accelerating from rest at 4 rad/s2 for 3 seconds.
- Worked exampleFind the time for a flywheel spinning at 30 rad/s to stop under a constant angular deceleration of 5 rad/s2.
- Common mistakeWhy doesn't a larger angular acceleration always mean a wheel is spinning faster right now?
Your turn1 of 2
A wheel starts from rest and accelerates at 4 rad/s2 for 6 s. Through what angle does it turn, in radians?Torque
- How it worksHow does torque depend on both the size of a force and exactly where and at what angle it's applied?
- Worked exampleFind the torque produced by a 50 N force applied perpendicular to a wrench 0.25 m from the bolt.
- Common mistakeWhy doesn't pushing harder on a door near its hinge open it as easily as pushing near the handle?
- Going furtherWhat happens to the torque a force produces as the angle between the force and the lever arm shrinks from 90 to 0 degrees?
Your turn1 of 2
A force of 62 N is applied perpendicular to a wrench at a distance of 0.36 m from the bolt. What torque does it produce, in newton-meters?Moment of Inertia by Integration
- How it worksHow do you integrate r squared dm across an object's mass distribution to find its moment of inertia?
- Worked exampleFind the moment of inertia of a uniform rod of mass 2 kg and length 1 m about its center, using I = (1/12) M L2.
- Worked exampleFind the moment of inertia of a uniform disk of mass 4 kg and radius 0.3 m rotating about its center.
- Common mistakeWhy doesn't moment of inertia depend on mass alone, the way ordinary inertia in F = ma does?
Your turn1 of 2
A uniform rod has mass 4 kg and length 1.47 m. What is its moment of inertia about its center, using I = (1/12) M L2?The Parallel-Axis Theorem
- How it worksHow does the parallel-axis theorem let you shift a moment of inertia from the center of mass to any other axis?
- Worked exampleFind the moment of inertia of a 2 kg rod of length 1 m about its end, given its center-of-mass value is (1/12) M L2.
- Going furtherWhat happens to an object's moment of inertia as the rotation axis moves farther from its center of mass?
- Common mistakeWhy isn't the moment of inertia about the center of mass ever larger than the moment of inertia about any parallel axis?
Your turn1 of 2
A rod of mass 4 kg and length 1.8 m has a moment of inertia about its center of (1/12) M L2. Using the parallel-axis theorem, what is its moment of inertia about an axis through one end, in kg m2?Newton's Second Law for Rotation
- How it worksHow does net torque equal moment of inertia times angular acceleration, mirroring F = ma exactly?
- Worked exampleFind the angular acceleration of a 0.5 kg m2 flywheel under a net torque of 2 N m.
- Common mistakeWhy doesn't the same torque produce the same angular acceleration on every object, regardless of its shape?
Your turn1 of 2
A net torque of 7 N m acts on a flywheel with moment of inertia 1.37 kg m2. What is its angular acceleration, in radians per second squared?Rotational Kinetic Energy and Work
- How it worksHow does one-half I omega squared play the same role for rotation that one-half m v squared plays for straight-line motion?
- Worked exampleFind the rotational kinetic energy of a 3 kg m2 wheel spinning at 10 rad/s.
- Ties togetherHow does the work-energy theorem for rotation mirror the same theorem you already learned for straight-line motion?
Your turn1 of 2
A wheel with moment of inertia 3.63 kg m2 spins at 18 rad/s. What is its rotational kinetic energy, in joules?Rolling Motion
- How it worksHow does rolling without slipping lock an object's linear speed to its angular speed through v = r omega?
- How it worksHow does splitting a rolling object's kinetic energy into translational and rotational parts explain why shape matters?
- Worked exampleFind the speed at the bottom of a 2 m high ramp for a solid sphere that rolls without slipping from rest.
- Common mistakeWhy doesn't a rolling object's kinetic energy come entirely from its motion across the ground?
Your turn1 of 2
A solid sphere rolls without slipping from rest down a ramp of height 2.24 m. Using energy conservation (g = 9.8 m/s2), what is its speed at the bottom, in meters per second?