Circular Motion and Angular Momentum Flashcards
What is the distance s a particle travels on a circular path equal to?
s = rϴ
Using the equation for distance in circular motion s, what is the equation for velocity?
v = ds/dt = r dϴ/dt = rω
What is the equation(s) for centripetal acceleration?
a = ωv = ω^2 r = v^2 / r
How can you make the equation v = rω vectorial?
Defining the angular velocity vector as a vector of length ω pointing along the axis of rotation. v-> = ω-> X r->, where -> denotes vector.
How do you find the velocity of something in orbit?
Use the equation for the gravitational force and set it equal to the mass multiplied by the centripetal acceleration.
What is a moment?
When you balance the product of weights times their distance from a pivot.
What is the equation for torque τ?
τ-> = r-> X F->
What is the angular momentum L->?
L-> = r-> X p->
What is the torque equal to in terms of momentum?
The rate of change of angular momentum. τ-> = dL-> / dt
What is the total torque from external forces on a composite body equal to?
The rate of change of its total angular momentum.
What is the torque of a body in equilibrium?
Zero.
How do you work out torque?
Multiply the distance from the pivot by the force. Include directions.
Where is the centre of gravity in terms of particles and torque?
The point at which the gravitational forces acting on the particles have zero total torque.
What is the equation for angular momentum of a particle travelling in a circle?
L = mrv = mr^2 ω
How do you derive the equation for angular momentum involving the moment of intertia I?
Have the kinetic energy of the particle and substitute in v = rω, and then set I = mr^2 (the moment of inertia). This can then be substituted into the angular momentum equation.
What is the general equation for the moment of inertia I?
I = integral of r^2 dm = integral of r^2 p dV.
What is the moment of inertia of a thin ring of radius R mass M about an axis through its centre, perpendicular to the plane of the ring?
I = MR^2
What is the moment of inertia of a uniform circular disc of radius R, mass M, about an axis through its centre, perpendicular to its face?
I = 1/2 MR^2
What is the moment of inertia for a uniform sphere, radius R, mass M?
I = 2/5 MR^2
What is the moment of inertia of a thin rod, length L, mass M, about a perpendicular axis through its centre?
I = 1/12 ML^2
What is the moment of inertia of a thin rod, length L, mass M, about a perpendicular axis through one end?
I = 1/3 ML^2
What is the parallel axis theorem?
The moment of inertia of an object around an axis that lies at a perpendicular distance d from the objects centre of mass equals the moment of inertia around a parallel axis through the centre of mass plus Md^2.
What is the equation for the angular velocity squared for a pendulum?
ω^2 = 4pi^2 / T^2 = 1/(1+(Icm/Md^2)) * (g/d)
Which laws can you use to prove the conservation of angular momentum and how?
Keplers laws of planetary motion. Second laws states that planets clear the same area in a set time no matter where they are on the orbital path. If you take an infinitesimally small dϴ, can think of a triangle and get equation for angular momentum - shows its conserved.
What is the kinetic energy when applying a force to spin up a rigid body?
T = 1/2 * I * ω^2 + 1/2 * m * v^2
What is the equation for work done when applying a force to spin up a rigid body?
dW = τ dθ
What is the equation for power when applying a force to spin up a rigid body?
P = dW/dt = τω
