Physics 207: Lecture 2 Notes

1
Lecture 11

• Goals
• Employ Newtons Laws in 2D problems with
  circular motion
• Relate Forces with acceleration

Assignment HW5, (Chapter 7, 8 and 9 due
10/18) For Wednesday Reading through 1st four
sections in Ch. 9
2
Exercise Tension revisited
Compare the strings below in settings (a) and (b)
and their tensions.

1. Ta ½ Tb
2. Ta 2 Tb
3. Ta Tb
4. Correct answer is not given

3
Chapter8 Reprisal of Uniform Circular Motion
For an object moving along a curved trajectory
with constant speed a ar (radial only)
4
Non-uniform Circular Motion
For an object moving along a curved trajectory,
with non-uniform speed a ar aT (radial and
tangential)
aT
ar
5
Uniform or non-uniform circular motion
aradial vT2 / r

• implies ?
• and if there is acceleration there MUST be a net
  force

6
Key steps

• Identify forces (i.e., a FBD)
• Identify axis of rotation
• Apply conditions (position, velocity
  acceleration)

7
Example The pendulum
axis of rotation
Consider a person on a swing
(A)
(B)
(C)
When is the tension equal to the weight of the
person swing? (A) At the top of the swing
(turnaround point) (B) Somewhere in the
middle (C) At the bottom of the swing (D)
Never, it is always greater than the weight
(E) Never, it is always less than the weight
8
Example Gravity, Normal Forces etc.
axis of rotation
at top of swing vT 0 Fr m 02 / r 0 T
mg cos q T mg cos q T lt mg
at bottom of swing vT is max Fr m ac m vT2
/ r T - mg T mg m vT2 / r T gt mg
9
Conical Pendulum (Not a simple pendulum)

• Swinging a ball on a string of length L around
  your head
• (r L sin q)

axis of rotation
S Fr mar T sin q S Fz 0 T cos q mg
so T mg / cos q (gt mg) mar (mg /
cos q ) (sin q ) ar g tan q vT2/r ?
vT (gr tan q)½
L
r
Period T 2p r / vT 2p (r cot q /g)½ 2p (L
cos q /g)½
10
Conical Pendulum (very different)

• Swinging a ball on a string of length L around
  your head

axis of rotation
Period t 2p r / vT 2p (r cot q /g)½ 2p
(L cos q / g )½ 2p (5 cos 5 / 9.8 )½
4.38 s 2p (5 cos 10 / 9.8 )½ 4.36 s 2p
(5 cos 15 / 9.8 )½ 4.32 s
11
Another example of circular motionLoop-the-loop
1
A match box car is going to do a loop-the-loop of
radius r. What must be its minimum speed vt at
the top so that it can manage the loop
successfully ?
12
Loop-the-loop 1
To navigate the top of the circle its tangential
velocity vT must be such that its centripetal
acceleration at least equals the force due to
gravity. At this point N, the normal force, goes
to zero (just touching).
Fr mar mg mvT2/r vT
(gr)1/2
vT
mg
13
Loop-the-loop 2
The match box car is going to do a loop-the-loop.
If the speed at the bottom is vB, what is the
normal force, N, at that point? Hint The
car is constrained to the track.
Fr mar mvB2/r N - mg N mvB2/r mg
N
v
mg
14
Orbiting satellites vT (gr)½
Net Force ma mg mvT2 / r
gr vT2 vT (gr)½The only difference
is that g is less because you are further from
the Earths center!
15
Geostationary orbit
16
Geostationary orbit

• The radius of the Earth is 6000 km but at 36000
  km you are 42000 km from the center of the
  earth.
• Fgravity is proportional to r -2 and so little g
  is now 10 m/s2 / 50
• vT (0.20 42000000)½ m/s 3000 m/s
• At 3000 m/s, period T 2p r / vT 2p 42000000
  / 3000 sec
• 90000 sec 90000 s/ 3600 s/hr 24 hrs
  
• Orbit affected by the moon and also the Earths
  mass is inhomogeneous (not perfectly
  geostationary)
• Great for communication satellites
• (1st pointed out by Arthur C. Clarke)

17
Home Exercise
Swinging around a ball on a rope in a nearly
horizontal circle over your head. Eventually the
rope breaks. If the rope breaks at 64 N, the
balls mass is 0.10 kg and the rope is 0.10
m How fast is the ball going when the rope
breaks? (neglect mg contribution, 1 N ltlt 40 N)
Fr m vT2 / r T vT (r Fr / m)1/2 vT
(0.10 x 64 / 0.10)1/2 m/s vT 8 m/s
18
Recap
Assignment HW5, For Wednesday Finish
reading through 1st four sections in Chapter 9