AP Physics: 06: Rotational Motion

Translation

if an imaginary line on an object stays parallel to its original position while the object moves, the motion of the object consists of a(n) ___

Rotation

if an imaginary line on an object does NOT stay parallel to its original position while the object moves, the motion of the object consists of a(n) ___

Rigid Body

for a rotating object, if all the points along a radial line have the same angular displacement (∆θ), then the object is referred to as a(n) ___

ω = ∆θ / ∆t

write the equation for average angular velocity

ω = average angular velocity

∆θ = angular displacement

∆t = change in time

ω = dθ / dt

write the derivative for instantaneous angular velocity

ω = average angular velocity

dθ = an infinitesimally small angular displacement

dt = an infinitesimally small change in time

α = ∆ω / ∆t

write the equation for average angular acceleration

α = average angular acceleration

∆ω = change average angular velocity

∆t = change in time

α = dω / dt

write the derivative for instantaneous angular acceleration

α = average angular acceleration

dω = an infinitesimally small change in angular velocity

dt = an infinitesimally small change in time

∆θ = ∆s / r

write the equation for angular displacement relative to arc length

∆θ = angular displacement

r = radius

∆s = change arc length

v = rω

write the equation for tangential (linear) velocity of a rotating object

v = tangential velocity

r = radius

ω = angular velocity

a = rα

write the equation for tangential (linear) acceleration of a rotating object

a = tangential acceleration

r = radius

α = angular acceleration

∆θ = ω∆t

write the equation for rotational motion with uniform angular acceleration, in which angular acceleration is not needed

∆θ = angular displacement

∆t = change in time

ωi = initial angular velocity

ωf = final angular velocity

ωf = ωi + α∆t

write the equation for rotational motion with uniform angular acceleration, in which angular displacement is not needed

∆t = change in time

ωi = initial angular velocity

ωf = final angular velocity

α = angular acceleration

∆θ = ωi + ½α(∆t)²

write the equation for rotational motion with uniform angular acceleration, in which final angular velocity is not needed

∆θ = angular displacement

∆t = change in time

ωi = initial angular velocity

α = angular acceleration

∆θ = ω∆t - ½α(∆t)²

write the equation for rotational motion with uniform angular acceleration, in which initial angular velocity is not needed

∆θ = angular displacement

∆t = change in time

ωf = final angular velocity

α = angular acceleration

ωf² = ωi² + 2α∆tθ

write the equation for rotational motion with uniform angular acceleration, in which change in time is not needed

∆θ = angular displacement

ωi = initial angular velocity

ωf = final angular velocity

α = angular acceleration

Torque(s)

the dynamics of translational motion involve describing the acceleration of an object in terms of its mass (inertia) and the forces that act on it; by analogy, the dynamics of rotational motion involve describing the angular (rotational) acceleration of an object in terms of its rotational inertia and the ___ that act on it

Torque

intuitively, ___ describes the effectiveness of a force in producing rotational acceralation

Torque, Angular (Acceleration)

just like a force is a vector quantity that produces linear acceleration, a(n) ___ is a vector quantity that produces ___

τ = rFsinθ

write the equation for torque

τ = torque

r = radius

F = force

θ = angle between force and radius

τ = lF

write the equation for torque using the line of action

τ = torque

l = lever arm (or moment arm) of the force relative to the pivot

F = force

Line of Action

an easier way to solve for torque: instead of determining the distance from the pivot point to the point of application of the force (which is the radius), find the perpendicular distance from the pivot point to what's called the ___ of the force, which is an infinite line representing the force; this distance is represented with the letter l

Lever Arm (or Moment Arm)

an easier way to solve for torque: instead of determining the distance from the pivot point to the point of application of the force (which is the radius), find the perpendicular distance from the pivot point to what's called the line of action of the force, which is an infinite line representing the force; this distance is called the ___ of the force relative to the pivot, and is denoted by l

τ = r┴F

concerning torque, since the "lever arm" is the component of the radius that's perpendicular to the force, it is also denoted by r┴; so the equation for torque can be written as:

τ = torque

F = force

r┴ = lever arm

Cross

torque is the (dot / cross) product of the radius and the force

Torque(s)

the net torque is the sum of all ___

mr²

for a point mass at a distance r from the axis of rotation, its rotational inertia (also called the moment of inertia) is defined as ___

I = ∫p(r)r² dV

for a continuous solid body, the sum of all the rotational inertias is represented by the integral:

I = total rotational inertia

r = radius

p(r) = mass density at each point r

dV = an infinitesimally small change in volume

τ = Iα

write the equation for net torque

τ = net torque

I = total rotational inertia

α = angular acceleration

K = ½Iω²

write the equation for rotational kinetic energy

K = rotational kinetic energy

I = total rotational inertia

ω = angular velocity

I = Icm + mr²

if the rotational inertia of a body is known relative to an axis that passes through the body's center of mass, then the rotational inertia, I, relative to any other rotation axis (as long as it is parallel) can be calculated by the following equation:

I = rotational inertia relative to another (but parallel) axis

Icm = rotational inertia relative to axis that passes through the center of mass

m = mass

r = perpendicular distance between the axis of rotation and the axis that would pass through the centre of mass

Parallel Axis

if the rotational inertia of a body is known relative to an axis that passes through the body's center of mass, then the rotational inertia, I, relative to any other rotation axis (as long as it is parallel) can be calculated by the following equation: I = Icm + mr²

this concept is known as the ___ theorem

At Rest

for a cylindrical solid rolling down an incline (and NOT slipping), the point of contact is instantaneouly ___

rω (Radius Times the Angular Velocity)

for a cylindrical solid rolling down an incline (and NOT slipping), the velocity, v, of the center of mass of the object is equal to:

W = τ∆θ

write the equation for work in terms of rotational motion

W = work

τ = torque

∆θ = angular displacement

W = ∫τ dθ

write the integral for work in terms of rotational motion

W = work

τ = torque

dθ = an infinitesimally small angular displacement

P = τω

write the equation for power in terms of rotational motion

P = power

τ = torque

ω = angular velocity

Angular Momentum

___ is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis

L = Iω

write the equation for angular momentum

L = angular momentum

I = total rotational inertia

ω = angular velocity

Crosss

angular momentum is the (dot / cross) product of radius and the linear momentum

τ = dL / dt

write the derivative for Newton's second law in terms of rotational motion

τ = net torque

dL = an infinitesimally small change in angular momentum

dt = an infinitesimally small change in time

Conservation of Angular Momentum

if a net force is zero, then the linear momentum must be constant; similarly, if a net torque is zero, then the angular momentum must be constant; this is known as the ___

Translational

an object is said to be in (translational / rotational) equilibrium if the sum of all the forces acting on it is zero

Rotational

an object is said to be in (translational / rotational) equilibrium if the sum of all the torques acting on it is zero

Static

if an object is at rest, then it is said to be in ___ equilobrium