1. Circular Motion – Part II

2. Goal of the class To understand the forces involved in circular motion
Question of the day: How does the tension in a string vary in circular motion?

3. Motion in a Horizontal Circle
e.g. on a frictionless table
A=v2/r from last time
Mg = N as no vertical acceleration
Fnet = ma = mv2/r towards the centre
The force is just the tension T = mv2/r

4. Conical Pendulum Speed v = constant Angle = theta radius r, Length L
Only 2 forces, mg and T
Resolve T into x and y
The pendulum acted on by 3 forces, mg down, Tcos theta up and T sin theta towards centre
As a_vert = 0 then F_vert = 0 as F=ma
This implies mg = Tcos theta and cancel out (1)
Fnet = Tsin theta = mv2/r (2)
Divide to eliminate T
Tan theta = v2/rg or v2 = rg tan theta

5. Going around the bend 1) Frictionless 2) Friction
Only forces are mg and N
But for circular motion a =v2/r This needs a force towards the centre that doesn’t exist
Therefore no circular motion on frictionless road
For friction fs =mus N
So fs = mv2/r or v2 = r. fs /m = r mus N / m = r mus g

6. Problem
A car rounds a banked round. The radius of curvature of the road is R, the banking angle is θ, and the coefficient of static friction is μ . Find the range of speeds the car may have without slipping up or down the bank
Don’t move x and y as you need v2/r in x direction
Resolve and factor out N
Sub 1 into 2 and find v2

7. Problem 2
A large vertical cylinder spins about its axis fast enough that any person inside is held against the wall when the floor drops away. The radius of the cylinder is R and the coefficient of static friction is μs. Find the maximum period of revolution required to stop the person from falling.
Minimum speed N=mv2/r
Mu_s mv2/r >= mg
V2 >= Rg/mu_s
V = root that
T = 2piR/V
Solve for T

8. Science Fair
I have a sign up sheet here for people who can sign up to the science fair if they wish
Friday 28th November

9. Extra Problem
A pail of water is spun in a vertical circle of radius 0.7m. What is the minimum speed at the top so that no water spills out?
For no water to come out it means N must be bigger than 0
Mv2/r > mg