Rotational Kinematics

Remember everything you learned way back when about linear (straight line) kinematics. It’s all the same except…

Rotation – circular motion of an object about an internal axis

Revolution – circular motion of object about an external axis

Angular Displacement – change in angular position from one point to another.

Δθ – angular displacement in radians, degrees, number

of rotations or revolutions, etc…

– positive for counter-clockwise (c.c.w.)

negative for clockwise (c.w.)

– dimensionless!

θ in radians (rad) = arclength / radius = s / r

– note units!

If s = r,

θ = s / r = r / r = s / s = 1 radian

An angle of 1 radian is swept out when the arc length (s) is equal to the radius (r) of the circle.

Rotational Kinematics (continued)

An angle of 360^o is swept out in one full circle. In one full circle, s = 2pr, so θ = s / r = 2pr / r = 2p radians!

360^o = 2p radians (Note, 2p ≈ 2*3.14 = 6.28)

57.3^o = 1 radian

1^o = 2p / 360 radians

Angular velocity – time rate of change in angular displacement

w = (θ_f - θ_i ) / t = Δθ / t = average angular velocity in rad/s

– note: rad/s = 1/s = s^-1

– positive for counter-clockwise (c.c.w.)

negative for clockwise (c.w.)

Angular acceleration – time rate of change in angular velocity

a = (w_f - w_i ) / t = Δw / t = average angular acceleration in rad/s²

– note: rad/s² = 1/s² = s^-2

– sign depends on …

Rotational Kinematics (continued)

All the other (linear) kinematics equations you know and love have a direct analog in rotational kinematics:

v_avg = Δx / Δt ω_avg = Δ θ / Δt

a = Δv / Δt α = Δ ω / Δt

v = v_o + a t ω = ω _o + α t

v_avg = ½ (v_o + v) ω _avg = ½ (ω _o + ω )

x = ½ (v_o + v) t θ = ½ (ω _o + ω ) t

x = v_o t + ½ a t² θ = ω_o t + ½ α t²

v² = v_o² + 2 a x ω² = ω _o ² + 2 α θ

Tangential Speed and Acceleration

w = θ / t θ = s / r s = r θ

tangential speed = v_T = x / t = s / t = r θ / t = r w

angular velocity w MUST BE IN rad/s

tangential acceleration = a_T = r a

angular acceleration a MUST BE IN rad/s²

Rotational Kinematics (continued)

Centripetal Acceleration Revisited

a_c = v_T² / r = (r w)² / r = r w²

Centripetal Acceleration PLUS Tangential Acceleration

If you have an object that is traveling in a circle but is either speeding up or slowing down (Δv_T ≠ 0, Δw ≠ 0), then this object has both a centripetal acceleration (a_c) AND tangential acceleration (a_T ≠ 0, a ≠ 0).

This object is in non-uniform circular motion.

The NET acceleration is the vector sum of a_c and a_T!

The “good” news is that a_c and a_T will always be mutually perpendicular, therefore, the vector sum will always just be a Pythagorean sum: a = SQRT(a_c² + a_T²)

Note that F_net will no longer = F_c (as it is in uniform circular motion.)

Rolling Motion

Consider the tire on a car as it moves down the street. In particular, consider the tire itself rolling along the street as the car moves down the street. Assume this tire makes one complete revolution in a time t.

If this tire rolls without slipping (and ALL our rolling-thing problems will be objects rolling without slipping), then in one complete rotation of the tire, the tire (and thus the car!) will have moved forward one circumference (of the tire.) Thus:

d = 2pr

v_car = d / t = 2pr / t = w r = v_T !!

The forward (linear) speed of the car is the same as the tangential speed of the edge of the tire about the axle.

This concept will be most important when combining linear kinetic energy with rotation kinetic energy (next chapter.)

Note that a similar analysis of acceleration gives:

a_car = r a = a_T

The forward (linear) acceleration of the car is the same as the tangential acceleration of the edge of the tire about the axle.