Fixed-axis Rotation Lesson 10 by Owen Borville 11.25.2025
The angular position θ of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
θ = s/r, where s is the arc length and r is the radius length.
The angular velocity of a rotating body about a fixed axis is ω (rad/s), the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body is ω = lim Δt->0 Δθ/ Δt = dθ/dt is the derivative with respect to time of the angular position θ, found by taking the limit lim Δt->0 in the average angular velocity ω = Δθ/ Δt
vt (tangential speed) = r ω (radius x angular velocity)
The angular velocity vector ω is found using the right hand rule: If the fingers curl in the direction of the rotation about a fixed axis, the thumb points in the direction of ω (vector).
Angular acceleration occurs when angular velocity is not constant. Average angular velocity over a given time is α = Δω/Δt.
The instantaneous angular acceleration is the time derivative of angular velocity α = lim Δt->0 Δω/ Δt = dω/dt.
If the rotation rate is decreasing, the angular acceleration is in the opposite direction to ω(angular velocity). If the rotation rate is increasing, the angular acceleration is in the same direction as angular velocity.
The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point. at = r α (tangential acceleration = radius x angular acceleration)
The kinematics of rotational motion describes the relationships among rotation angle (angular position), angular velocity, angular acceleration, and time.
Average angular velocity is ω = (ωo + ωf)/2
Angular displacement is the change in an object's angular position as it rotates around a fixed axis = θf = θo + ω(avg)t
Angular velocity from constant angular acceleration = ωf = ωo + αt
Angular velocity from displacement and constant angular acceleration = θf = θo + ωot + 1/2 αt^2
Change in angular velocity = ωf^2 = ωo^2 + 2 α (Δθ)
Total linear acceleration (vector) = αc + αt
Rotational kinetic energy is the energy an object possesses due to its rotation around an axis = K = 1/2(∑ mj rj^2) ω^2
Moment of Inertia is a measure of an object's resistance to rotational acceleration (or rotational inertia) = I = ∑ = mj rj^2
Rotational kinetic energy (moment of inertia) = K + 1/2 I ω^2
Moment of Inertia (of a continuous object) = I ∫ r^2 dm
The parallel axis theorem states that the moment of inertia (I) about any axis is equal to the moment of inertia about a parallel axis through the center of mass (Icm) plus the product of the object's mass and the square of the distance between the two axes.
Parallel axis theorem = I (parallel axis) = I (center of mass) + md^2
Moment of inertia of a compound object is found by adding the individual moments of inertia of its component parts= I(total) = ∑ Ii
Torque is a force that causes an object to turn or rotate, or the power of such a force. Torque is the rotational correspondent of linear force (moment of force).
Torque vector = 𝜏 = r * F is the rotational force on an object, having both magnitude and direction.
Magnitude of Torque = |𝜏 | = r ⟂ F = 𝜏 rFsin θ
The direction of the torque vector is perpendicular to both the position and force vectors.
Total torque = 𝜏net = ∑ |𝜏i|
Newton 's Second Law for Rotation = ∑𝜏i = I α (moment of inertia * angular acceleration)
Incremental (infinitesimal) work done by Torque = dW = (∑𝜏i) dθ (infinitesimal angular displacement)
Constant work done by torque = W = 𝜏Δθ
Work-Energy Theorem states that the net work done on an object equals the change in its kinetic energy= Wab = Kb - Ka
Rotational work done by net force = Wab = ∫(θa,θb)(∑𝜏i) dθ
Rotational Power is the rate at which rotational work is done (which is torque times angular velocity)= P = 𝜏ω
The angular position θ of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
θ = s/r, where s is the arc length and r is the radius length.
The angular velocity of a rotating body about a fixed axis is ω (rad/s), the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body is ω = lim Δt->0 Δθ/ Δt = dθ/dt is the derivative with respect to time of the angular position θ, found by taking the limit lim Δt->0 in the average angular velocity ω = Δθ/ Δt
vt (tangential speed) = r ω (radius x angular velocity)
The angular velocity vector ω is found using the right hand rule: If the fingers curl in the direction of the rotation about a fixed axis, the thumb points in the direction of ω (vector).
Angular acceleration occurs when angular velocity is not constant. Average angular velocity over a given time is α = Δω/Δt.
The instantaneous angular acceleration is the time derivative of angular velocity α = lim Δt->0 Δω/ Δt = dω/dt.
If the rotation rate is decreasing, the angular acceleration is in the opposite direction to ω(angular velocity). If the rotation rate is increasing, the angular acceleration is in the same direction as angular velocity.
The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point. at = r α (tangential acceleration = radius x angular acceleration)
The kinematics of rotational motion describes the relationships among rotation angle (angular position), angular velocity, angular acceleration, and time.
Average angular velocity is ω = (ωo + ωf)/2
Angular displacement is the change in an object's angular position as it rotates around a fixed axis = θf = θo + ω(avg)t
Angular velocity from constant angular acceleration = ωf = ωo + αt
Angular velocity from displacement and constant angular acceleration = θf = θo + ωot + 1/2 αt^2
Change in angular velocity = ωf^2 = ωo^2 + 2 α (Δθ)
Total linear acceleration (vector) = αc + αt
Rotational kinetic energy is the energy an object possesses due to its rotation around an axis = K = 1/2(∑ mj rj^2) ω^2
Moment of Inertia is a measure of an object's resistance to rotational acceleration (or rotational inertia) = I = ∑ = mj rj^2
Rotational kinetic energy (moment of inertia) = K + 1/2 I ω^2
Moment of Inertia (of a continuous object) = I ∫ r^2 dm
The parallel axis theorem states that the moment of inertia (I) about any axis is equal to the moment of inertia about a parallel axis through the center of mass (Icm) plus the product of the object's mass and the square of the distance between the two axes.
Parallel axis theorem = I (parallel axis) = I (center of mass) + md^2
Moment of inertia of a compound object is found by adding the individual moments of inertia of its component parts= I(total) = ∑ Ii
Torque is a force that causes an object to turn or rotate, or the power of such a force. Torque is the rotational correspondent of linear force (moment of force).
Torque vector = 𝜏 = r * F is the rotational force on an object, having both magnitude and direction.
Magnitude of Torque = |𝜏 | = r ⟂ F = 𝜏 rFsin θ
The direction of the torque vector is perpendicular to both the position and force vectors.
Total torque = 𝜏net = ∑ |𝜏i|
Newton 's Second Law for Rotation = ∑𝜏i = I α (moment of inertia * angular acceleration)
Incremental (infinitesimal) work done by Torque = dW = (∑𝜏i) dθ (infinitesimal angular displacement)
Constant work done by torque = W = 𝜏Δθ
Work-Energy Theorem states that the net work done on an object equals the change in its kinetic energy= Wab = Kb - Ka
Rotational work done by net force = Wab = ∫(θa,θb)(∑𝜏i) dθ
Rotational Power is the rate at which rotational work is done (which is torque times angular velocity)= P = 𝜏ω