Reminders:

NO SI THURSDAY!!
Exam Overview
“Approximately 1/3 of the problems will
stress understanding of the physics
concepts, whereas the remainder will be
numerical problems to test ability to apply
these concepts.” -Syllabus
27 Questions
Statistical Breakdown of Exam 2
26% Energy
 17% Collisions (+Momentum)
 46% Rotational (Center of mass, moment of
inertia, statics, angular momentum, torque)
 10% Gravitational

Let’s put some analogies to work:
Not knowing gravitation would be like losing 3 fingers
Not knowing rotational is like losing a heart or both
lungs
Energy

Conservation of Energy
 Conservative/Non-conservative
 Potential Energy
Springs
 Collisions
 Rotational
 Gravitational Force/PE

KE

2 types


1
Rotational 𝐾𝐸 = 𝑚𝑣 2
2
1
Translational 𝐾𝐸 = 𝐼𝜔2
2
Springs

A 10 kg mass hits a spring at a speed of
50 m/s. The spring has k=30 N/m.
 How far will the spring be compressed?
 What will the PE of the spring be when it is
fully compressed?
 What will the PE of the spring be when it is
halfway compressed?



Conservation
of
Energy
Find a method that works for you.
What’s most important is that you fully understand your method, front
and back.
For me, I prefer doing one of two things:
 If there’s no friction, I just write out the energy at the two points.
 If there is friction, I use
𝑊𝑜𝑡ℎ𝑒𝑟 = Δ𝑃𝐸 + Δ𝐾𝐸
 𝑊𝑜𝑡ℎ𝑒𝑟 =Work done by forces other than gravity

Define Mechanical Energy
1. KE
2. PE
3. KE+PE
Given a PE graph or function,
what is F?





𝑈 is just another
symbol for 𝑃𝐸
Where are the stable
and unstable
equilibriums?
At what points is the
force zero?
At what points is the
force negative?
Where does the
particle have
maximum speed, if it’s
released at x=4?
𝑑𝑈
−
𝑑𝑥

𝐹=

Recall that at equilibriums, the force is
𝑑𝑈
zero. AKA
=0
𝑑𝑥
You’re given 𝑈 = 20𝑥 3
 What is F at x=-2?
Conservative Forces vs.
Non-conservative Forces

Conservative
 Always associated with some PE
 Work doesn’t depend on path taken
 Examples: Gravity, Electrostatic

Not Conservative
 PE doesn’t exist
 Work does depend on path
 Examples: Friction, Air resistance
Collisions
𝑷 = 𝑚𝒗: Momentum is a vector
 Conservation of momentum
 Impulse 𝑱 = 𝚫𝐏
 Elastic, Inelastic, Completely Inelastic

Deciding elasticity
Do they tell you it’s
elastic, inelastic, or
completely inelastic?
Yes
Does the problem say
the objects stick
together after the
collision?
No
Yes
No
It’s completely
inelastic (KE not
conserved)
What is the Δ𝐾𝐸 of
the collision?
0
<0
Elastic
Inelastic (could be
completely inelastic,
but doesn’t have to
be)
Is Momentum Conserved in a
Collision

In this course, YES, ALWAYS!
Impulse

Impulse J: 3 ways to define
 𝑱 = Δ𝐏
 𝑱 = ∫ 𝑭𝑑𝑡
 𝑱 = 𝐹Δ𝑡 𝑓𝑜𝑟 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑓𝑜𝑟𝑐𝑒

The impulse of A on B is equal and opposite to the
impulse of B on A

A 5 kg ball hits a wall at 8 m/s and bounces back at
the same speed. If the collision took 4 seconds,
what is the average force done by the wall on the
ball?
Rotational

Moment of Inertia=Rotational Inertia:
 𝐼 = 𝑚𝑖 𝑟𝑖2 (about some axis)

Torque
 𝜏 = 𝒓 × 𝑭 (about some axis)
 𝜏 = 𝛼𝐼 (about some axis)

Angular momentum
 𝑳 = 𝒓 × 𝒑 = 𝒓 × 𝑚𝒗 (about some axis)
Notice how everything angular is about some
axis. Make sure that your choice of axes match.
 Center of Mass!!

Center of Mass!!

Why is this important/grouped under
rotation?
=
Solve…

Mathematically…

Intuitively
 Do circular or non-circular go faster down
hills?
Angular momentum is the same for the two
systems below about an axis through O.
Discrete/Continuous…which
formula should I use?
Discrete: 𝐼 = ∑𝑚𝑖 𝑟𝑖2
Discrete: 𝐋 = ∑𝒓 × 𝑚𝒗
Continuous: 𝐼 = ∫ 𝑟 2 𝑑𝑚
(use equation sheet for this)
Continuous: 𝐋 = 𝐼𝝎
Worked example
What is the
angular
momentum?
Parallel Axis Theorem (PAT): 𝐼 =
2
𝐼𝑐𝑚 + 𝑀𝑑
Torque (𝜏 = 𝑟 × 𝐹 = 𝐼𝛼)

If net torque is zero, an object can still
be moving.
 In fact, the object can even be accelerating!
 However, the angular acceleration must be
zero, and the angular velocity must be
constant.
Comparison
between linear and
angular
Notice that there’s always
an axis involved with every
single rotational equivalent!
𝒙 (Position)
𝜃 (Angular position)
𝒗 (Linear velocity)
𝝎 (Angular velocity)
𝒂 (Linear acceleration)
𝜶 (Angular acceleration)
𝑚 (𝑴𝒂𝒔𝒔)
𝑭 = 𝑚𝒗 (Equation for force)
I (Moment of Inertia)
𝝉 = 𝐼𝝎 (Equation for torque)
𝝉 = 𝒓 × 𝑭 (Equation relating force and torque)
𝑳 = 𝒓 × 𝒑 (Definition of angular momentum)
𝐏 = m𝐯 (Definition of linear momentum)
1
𝐾𝐸𝑡𝑟𝑎𝑛𝑠 = 2 𝑚𝑣 2 (Definition of translational KE)
𝑭=
𝑑𝒑
𝑑𝑡
(Force-momentum relation)
1
𝐾𝐸𝑟𝑜𝑡 = 2 𝐼𝜔2 (Definition of rotational KE)
𝑭=
𝑑𝒑
𝑑𝑡
(Torque-angular momentum relation)
𝚫𝐏 = ∫ 𝐅 ⋅ dt (Impulse equation)
𝚫𝐋 = ∫ 𝝉 ⋅ dt (Angular Impulse equation)
𝑃 = 𝐹𝑣 (Power and constant force, constant velocity
relation)
𝑊 = ∫ F dx (Work caused by force)
P = 𝜏𝜔 (Power and constant torque, constant
angular velocity relation)
𝑊 = ∫ 𝜏 𝑑𝜃 (Work caused by torque)
Statics
Can be solved in 2 minutes
 Strategy: Generally, use conservation of
angular momentum first!

Both discrete and continuous!
Is Momentum Conserved?
 Is Angular Momentum Conserved?

Gravitation



𝐹=
𝐺𝑚𝑀
𝑟2
𝑃𝐸𝑔 =
𝐺𝑚𝑀
−
𝑟
𝑇 = 2𝜋
𝑟3
𝐺𝑀
Equation relating the
period of a planet’s motion around a star
of mass M
 NOTE: r is the distance between the two
masses, M is the mass of the object being
orbited.
Examples:
A planet is in a circular orbit around a star. The
mass of the star is 5 * 1028 kg. If the period of the
planet’s orbit is 1.00 * 105 s, then the orbital
radius of the planet around the star is ____ m.
The centers of two small uniform spherical
bodies are separated by a distance d and the
magnitude of the attractive gravitational force of
one on the other is F. If the distance between the
centers of the bodies decreases to d/2, the
magnitude of the force of one on the other
becomes ____ .
A 1.53-kg mass hangs on a rope
wrapped around a disk pulley of
mass 7.07 kg and radius 66.0 cm.
The rope does not slip on the
pulley.
 What is the angular acceleration
of the pulley?

If the block has fallen 0.8 m,
what is the speed of the block at
that time? (Two ways to solve
this)
Other Concepts

Magnitude of Forces, vectors, etc…
Multi-Principle Problems
A
Trickier Problems