PHYS1001 Lecture Notes - Lecture 9: Spinning Wheel, Maryland Route 2, Cross Product
dt
r
2
L
2
2
0
Rotational Motion
Angular displacement: An angle in radians which describes a rotation around some axis.
Angular velocity: Change in angle over time. The angle is measured from the positive X axis
Formulae:
dθ
dt
Tangential velocity:
- v = ds = r dθ = rω
dt dt
Rotational acceleration
- α = dω (rad/s2)
- The acceleration vector has 2 components, tangential and centripetal. These are at right
angles. Just add the vectors. acentripetal
= v2 = rω2
- atangential = dv = r dw = rα
dt dt
Kinetic energy in rotation
- Like asteroids orbiting around the sun.
- For a particle in a system, where all the particles have the same ω , Ki = 1mi(riω)2
- KE of a body is:
- ∑ 1mir2ω2 = 1Iω2
2 i 2
- I is the moment of inertia
- You can think of moment of inertia as being analogous to mass, making
something more difficult to accelerate.
- For the ballerina effect
- K1I1 = K2I2
Moment of inertia
- Analogous to mass.
- I = ∫r2dm = ∫r2ρdV
- ⍴ is the mass per unit length of an object
- ρ = M
Some rotational motion equations are analogous to linear equations
- d2θ = α
dt
- ω = ω0 + αt
- θ = θ0 + ω0t + 1αt2
- ω2 = ω2 + 2α(θ − θ0)
Rigid bodies
- A rigid body doesn’t deform when a force is applied to it. Maybe like a steel disc
ω =
s
- The distance between two points on the body doesn’t change no matter how
much force is applied to it. → Like a lot of things, only theoretical.
- Similarly, while a circular object may be undergoing motion of some kind, we
can’t represent it as a point like we could with linear and planar mechanics.
Angular velocity and acceleration
- Rotational motion can be combined with traditional motion as the motion of the axis of
rotation.
- To determine the axis of rotation, we employ the right hand rule in terms of the direction
of rotation to find the positive direction for the axis of rotation.
- It’s best to describe rotation in terms of some line OP where P is a fixed point on the
rotating body.
- Now we only really need the angle between the X-axis and the line as well as
time to describe all of rotational motion.
- The angle is measured in radians, which is a ratio of distances which is why it is
dimensionless, and the FAR SUPERIOR way of measuring angles.
- Angular Velocity
- The time-rate of change of the angle the line OP forms with the X-axis.
- The unit is radians per second, or s−1
- This has the same formula as linear velocity, but with angle instead of distance.
- Idk why, but we say this velocity is in the z-direction.
- “Angular velocity” refers to instantaneous angular velocity. You have to be more
specific if you want a different type like average angular velocity.
- Remember:
- 1 rpm =
2π −1
60
- 1 rps = 2π s−1
- Angular Acceleration
- The time-rate of change of the angular velocity of the line OP
Linear Speed in rotational motion
- For example, when we have to describe the motion of a single particle (What is its
tangential velocity).
- Since s = rθ , using the speed equation
- |
ds | = r|
dθ| = r|ω| = |vtan| → since speed is always positive.
dt dt
- This is equal to the instantaneous linear speed of the particle, but not the
average speed or average tangential velocity.
Linear Acceleration
- I should add, linear speed and acceleration are only linear in an instantaneous sense
rather than over real time.
- Tangential acceleration of a particle refers to a change in the magnitude of tangential
velocity
- Centripetal acceleration of a particle refers to a change in direction of tangential velocity
a is the acceleration in m/s2 as a result of the tangential and radial accelerations (as these are also
in m/s2
Document Summary
Angular displacement: an angle in radians which describes a rotation around some axis. The angle is measured from the positive x axis. Tangential velocity: d dt v = ds = r d = r dt dt. The acceleration vector has 2 components, tangential and centripetal. Just add the vectors. acentripetal atangential = dv = r dw = r r. For a particle in a system, where all the particles have the same , ki = 1mi(ri )2. You can think of moment of inertia as being analogous to mass, making something more difficult to accelerate. Is the mass per unit length of an object. Some rotational motion equations are analogous to linear equations d2 = . 2 = 2 + 2 ( 0) A rigid body doesn"t deform when a force is applied to it. The distance between two points on the body doesn"t change no matter how much force is applied to it.
