Angular Momentum Rotational Motion Lesson 11 by Owen Borville 11.26.2025
Uniform circular motion is the motion with a constant angular velocity ω = dθ/dt
Angular acceleration is non-uniform circular motion where the velocity changes with time and the rate of change of angular velocity: α = Δω/Δt.
Linear or tangential acceleration is the changes in the magnitude of velocity but not its direction at = Δv/Δt
For circular motion, v = rω and at = Δ(rω)/Δt
Δ(rω) = rΔω
at = rΔω/Δt
α = Δω/Δt
at = rα
α = αt/r
Rotational motion includes the relationships between rotation angle, angular velocity, angular acceleration, and time.
Average angular velocity ω(avg) = (ω0 + ω)/2
Average velocity = (v0 + v)/2
Angular acceleration increases the farther the force is applied from the pivot.
Angular acceleration is inversely proportional to mass.
If F (force) = ma (mass times acceleration) , and a(acceleration) = rα(radius times angular velocity), then F = mrα
Torque is the turning effectiveness of a force and the force that causes an object to rotate about an axis. Since F is perpendicular to r, torque is 𝜏 = rF
If rF = mr^2 α, then
𝜏 = mr^2 α
Rolling motion without slipping includes a static friction force between the rolling object and the surface. The linear velocity, acceleration, and distance of the center of mass are multiplied by the radius of the object to find velocity, acceleration, and distance of the object.
vCM = Rω
aCM = Rα
dCM = Rθ
Rolling motion with slipping would create a kinetic friction force between the rolling object and the surface. Energy is conserved in rolling motion without slipping, but not conserved with slipping.
Moment of inertia (I) of an object is the resistance to changes in an object's rotational motion and is the sum (∑) of mr^2 = I = ∑ mr^2
Torque is equal to the moment of inertia divided by angular acceleration: 𝜏 = Iα
α = net 𝜏/I or angular acceleration is equal to the net torque divided by the moment of inertia.
Rotational Kinetic Energy for an object (the energy an object possesses due to its rotation around an axis) with a moment of inertia I and an angular velocity ω =
KErot = 1/2 I ω^2
Helicopters store large amounts of rotational kinetic energy in their blades and use this stored energy to take off and during the flight. Also flywheels use stored rotational kinetic energy, along with the wheels of automobiles and bicycles.
The fundamental principles of work and energy are the same for both linear and rotational motion.
The work-energy theorem states that for rotational motion,
the net W = 1/2 Iωf^2 - 1/2 Iωi^2 (The net work is 1/2 the moment of inertia times the final angular velocity squared minus 1/2 the moment of inertia times the initial angular velocity squared).
Angular momentum (L) is the rotational equivalent of linear momentum representing an object's tendency to keep spinning.
Angular momentum = L = Iω (moment of inertia times the angular velocity.
The angular momentum vector l = r*p of a single particle is the product of the position vector r and the linear momentum. The angular momentum of a system of particles about a designated origin is the vector sum of the individual momenta of the particles in the system.
Torque (𝜏) = ΔL/Δt (change in angular momentum divided by the change in time)
Angular momentum is conserved, just like energy and linear momentum (law of conservation of angular momentum). Angular momentum is conserved when the net external torque is zero (just as linear momentum is conserved when the net external force is zero).
L = Iω along the axis of rotation is the angular momentum of a rotating rigid body.
Net Torque (𝜏) = ΔL/Δt , therefore angular momentum can be changed by torque. The net torque on a system is also the time derivative of its total angular momentum:
∑𝜏(net torque) = dL/dt directed along the axis of rotation.
Angular momentum (L) = constant with net 𝜏 = 0. Therefore, in conservation of angular momentum, angular momentum is conserved if it is constant with no net external torque and there is an inverse relationship with inertia and velocity. Angular momentum may be conserved in collisions, when no net external torque acts on the system.
The angular momentum of a system of particles is the sum of the individual momentum of each particle: L = l1 + l2 + ...lN This also satisfies the conservation of angular momentum. The derivative of total angular momentum also equals torque for a system of particles.
Angular velocity (ω) is inversely proportional to the moment of inertia (I) when total angular momentum (L) is conserved. Ifinal * ωfinal = Iinitial * ωinitial
Torque is perpendicular to the plane formed by r and F and is the direction of one's right thumb would point if fingers of the right hand are curled in the direction of F. The direction of torque is equal to the angular momentum produced.
A gyroscope is a device containing a rapidly spinning wheel that is used to detect orientation, angular velocity, and the deviation of an object from its desired orientation. A gyroscope works by using the conservation of angular momentum, which causes a spinning rotor to resist changes in its axis of rotation. A spinning wheel is attached to a series of rings that allow the inner wheel to maintain its orientation in space, regardless of how the outer frame moves. Resisting changes in orientation gives the gyroscope its stability.
Earth behaves as a gyroscope, with its angular momentum along its axis and pointing toward the North Star.
Precessional angular velocity is the rate at which the spin axis of a rotating object, such as a gyroscope, sweeps out a cone around a central axis. ωp = rMg/Iω
Uniform circular motion is the motion with a constant angular velocity ω = dθ/dt
Angular acceleration is non-uniform circular motion where the velocity changes with time and the rate of change of angular velocity: α = Δω/Δt.
Linear or tangential acceleration is the changes in the magnitude of velocity but not its direction at = Δv/Δt
For circular motion, v = rω and at = Δ(rω)/Δt
Δ(rω) = rΔω
at = rΔω/Δt
α = Δω/Δt
at = rα
α = αt/r
Rotational motion includes the relationships between rotation angle, angular velocity, angular acceleration, and time.
Average angular velocity ω(avg) = (ω0 + ω)/2
Average velocity = (v0 + v)/2
Angular acceleration increases the farther the force is applied from the pivot.
Angular acceleration is inversely proportional to mass.
If F (force) = ma (mass times acceleration) , and a(acceleration) = rα(radius times angular velocity), then F = mrα
Torque is the turning effectiveness of a force and the force that causes an object to rotate about an axis. Since F is perpendicular to r, torque is 𝜏 = rF
If rF = mr^2 α, then
𝜏 = mr^2 α
Rolling motion without slipping includes a static friction force between the rolling object and the surface. The linear velocity, acceleration, and distance of the center of mass are multiplied by the radius of the object to find velocity, acceleration, and distance of the object.
vCM = Rω
aCM = Rα
dCM = Rθ
Rolling motion with slipping would create a kinetic friction force between the rolling object and the surface. Energy is conserved in rolling motion without slipping, but not conserved with slipping.
Moment of inertia (I) of an object is the resistance to changes in an object's rotational motion and is the sum (∑) of mr^2 = I = ∑ mr^2
Torque is equal to the moment of inertia divided by angular acceleration: 𝜏 = Iα
α = net 𝜏/I or angular acceleration is equal to the net torque divided by the moment of inertia.
Rotational Kinetic Energy for an object (the energy an object possesses due to its rotation around an axis) with a moment of inertia I and an angular velocity ω =
KErot = 1/2 I ω^2
Helicopters store large amounts of rotational kinetic energy in their blades and use this stored energy to take off and during the flight. Also flywheels use stored rotational kinetic energy, along with the wheels of automobiles and bicycles.
The fundamental principles of work and energy are the same for both linear and rotational motion.
The work-energy theorem states that for rotational motion,
the net W = 1/2 Iωf^2 - 1/2 Iωi^2 (The net work is 1/2 the moment of inertia times the final angular velocity squared minus 1/2 the moment of inertia times the initial angular velocity squared).
Angular momentum (L) is the rotational equivalent of linear momentum representing an object's tendency to keep spinning.
Angular momentum = L = Iω (moment of inertia times the angular velocity.
The angular momentum vector l = r*p of a single particle is the product of the position vector r and the linear momentum. The angular momentum of a system of particles about a designated origin is the vector sum of the individual momenta of the particles in the system.
Torque (𝜏) = ΔL/Δt (change in angular momentum divided by the change in time)
Angular momentum is conserved, just like energy and linear momentum (law of conservation of angular momentum). Angular momentum is conserved when the net external torque is zero (just as linear momentum is conserved when the net external force is zero).
L = Iω along the axis of rotation is the angular momentum of a rotating rigid body.
Net Torque (𝜏) = ΔL/Δt , therefore angular momentum can be changed by torque. The net torque on a system is also the time derivative of its total angular momentum:
∑𝜏(net torque) = dL/dt directed along the axis of rotation.
Angular momentum (L) = constant with net 𝜏 = 0. Therefore, in conservation of angular momentum, angular momentum is conserved if it is constant with no net external torque and there is an inverse relationship with inertia and velocity. Angular momentum may be conserved in collisions, when no net external torque acts on the system.
The angular momentum of a system of particles is the sum of the individual momentum of each particle: L = l1 + l2 + ...lN This also satisfies the conservation of angular momentum. The derivative of total angular momentum also equals torque for a system of particles.
Angular velocity (ω) is inversely proportional to the moment of inertia (I) when total angular momentum (L) is conserved. Ifinal * ωfinal = Iinitial * ωinitial
Torque is perpendicular to the plane formed by r and F and is the direction of one's right thumb would point if fingers of the right hand are curled in the direction of F. The direction of torque is equal to the angular momentum produced.
A gyroscope is a device containing a rapidly spinning wheel that is used to detect orientation, angular velocity, and the deviation of an object from its desired orientation. A gyroscope works by using the conservation of angular momentum, which causes a spinning rotor to resist changes in its axis of rotation. A spinning wheel is attached to a series of rings that allow the inner wheel to maintain its orientation in space, regardless of how the outer frame moves. Resisting changes in orientation gives the gyroscope its stability.
Earth behaves as a gyroscope, with its angular momentum along its axis and pointing toward the North Star.
Precessional angular velocity is the rate at which the spin axis of a rotating object, such as a gyroscope, sweeps out a cone around a central axis. ωp = rMg/Iω
