Download PES 1110 Fall 2013, Spendier Lecture 10/Page 1 Today:

PES 1110 Fall 2013, Spendier
Lecture 10/Page 1
Today:
- Newton’s 2nd law
- HW due today
- Seminar announcement: Please join us for a very interesting seminar on a
telescope that uses a thin plastic membrane as a lens. Called a photon sieve, it works by
bending light through billions of tiny holes in the "cling film" style optic. Unlike in
existing traditional mirror-based telescopes, the new lens can be folded to fit into a tiny
space. 11am, Friday September 20, Room A204 Osborne
- Will work hard to get practice exam out by Friday!
- Friday we will have 2nd Quiz!
Recap Monday:
Forces to be identified in any problem:
- Weight - w , the downward force on an object due to gravity.
- Normal Force - Fn , the perpendicular force exerted by one solid
object onto another solid object.
- Friction - f , force which slows a moving object, always
opposed to the motion, 180 degree away from velocity) opposite to v .
- Tension - T , pulling force exerted by rope, chain, or spring, always at same angle as
rope. If you have a rope at 30 degrees in a problem the tension vector will have the same
angle.
Lots of components of vectors and vector addition.
More on weight:
mass
SI
kg
British
slug
English Engineering lbm
(pound mass)
a
m/s2
ft/s2
ft/s2
force (weight)
m kg/s2 = Newton
slug ft/ s2 = pounds
lbm ft/ s2 = lbf (pound force)
How do we measure forces?
We can use a spring balance: The distance x which the object moves due to the object's
weight is proportional to the force.
(We will learn later: F = kx, k is a constant describing the stiffness of the spring)
PES 1110 Fall 2013, Spendier
Lecture 10/Page 2
Newton’s First Law: The Law of Inertia
An object at rest stays at rest, an object in uniform motion stays in uniform motion if (and
only if) the net force acting on the object is zero.
∑F =0
net
Uniform motion - Straight line and constant speed, i.e, constant velocity.
Inertia - The property of all matter to stay in motion if already in motion; to stay at rest if
already at rest.
Inertial Reference Frames
Reference frames in which Newtonian mechanics holds are called inertial reference
frames or inertial frames. These are reference frames that move with constant velocity ,
including v = 0. We typically say that the ground is an inertial frame. But we know from
last lecture that the earth’s rotation causes a small centripetal acceleration. We calculated
it to be 0.3% of g and hence we will ignore it and treat the earth surface as an inertial
frame.
Said something stupid last time: a hanging object has two forces (tension and weight)
acting on it – the only problem where there is only one force is a free-falling object!
First Law Example 1)
A 5 kg mass is hung from the ceiling using a "massless" rope. What is the magnitude of
the tension force exerted by the rope on the mass? Hint: A 5 kg mass has a weight of 49N
on earth where this problem is taking place.
Answer: 49 N
First Law Example 2)
Two 5 kg masses are connected to each over pulleys using a rope. What is the tension
force that the rope exerts on the right-hand mass if they are both at rest?
Answer: 49 N
PES 1110 Fall 2013, Spendier
Lecture 10/Page 3
Newton’s Second Law:
This is only the really equation we will be using in chapters 5 and 6. We will be coming
back to this in a big way in chapter 6. Right now we will give a brief introduction and
then practice hard using it. We will see that we don’t really need a first law, we talked
about it for historical reasons, since the first law is contained in the second law.
F
∑ = 0 then we have a constant velocity
net
dv Constant v = constant ⇒
=a=0
dt
So if the forces do not add up to zero – what does it tell us?
The first law tells us that if
∑F ≠0⇒a≠0
net
Then the acceleration is also not zero. So the first law is the case of zero acceleration that
the second law itself contains.
So the big thing is that: Forces cause acceleration
This is the physical effect – when the forces do not add up to zero the acceleration is nonzero. This is why we had to do kinematics first because it would not have made that much
sense if you did not understand what acceleration is.
This is how nature works, from the bottom up, forces cause acceleration, the acceleration
in turn determines the velocity, and then acceleration and velocity together determine the
position. We learned it the other way position-velocity-acceleration-forces
Sir Newton found more than that. He found that the acceleration is:
(a) In the same direction as the net force
We learned that acceleration vectors are tricky things. They can change both speed and
direction. So to help you out it is a good thing to remember that when an object starts
from rest, the direction in which an object moves is in the direction of its acceleration. If I
push an object from rest to the right the net force will be to the right and it will accelerate
to the right.
(b) (the amount of acceleration is) Directly proportional to the net force
So I exert some number of Newton’s to the cart it accelerates. If I double the amount of
force I exert, apply twice as much force, it will give me twice as much acceleration.
(c) Inversely proportional to the mass
I can push a block for you. Can I push the wall? I cannot make the wall accelerate. The
wall is too big. How do we measure how big the wall is? We throw in the “mass” with
the more mass the smaller the acceleration. Mass is the amount of matter inside an object.
PES 1110 Fall 2013, Spendier
Lecture 10/Page 4
It is the number of protons and neutrons inside of an object, that determine mass. More
mass for a given force the smaller the acceleration.
Putting these three things together you get:
Second Law of Motion:
∑F
net
m
=a ⇒ ∑ F = ma
net
 kg i m 
units  2  = [ N ]...Newton
 s 
The net force is equal the mass times acceleration! (Don't say F = ma, here F stands
for single force!)
Here we stress that it is not just one force which determined the acceleration, it is all of
the forces together, it is the net force. It is the vector addition of the forces which equals
mass times acceleration.
Be aware, to apply the law we have to use mass in kg. So in problems you need to make
sure that the mass is given in kg – otherwise you need to convert!
Second Law Examples
Example 1: A 6860N car is in free-fall, what it its mass?
An object in free-fall has the only force acting on it – it is gravity – this is the world
famous single force problem! Object in free fall has acceleration 9.8 m/s2
w = m g (weight and acceleration are downward)
m = W/g = 6860/9.8 = 700 kg
Example 2: A 6860N car is sitting stationary on the ground, what is its mass?
It will have the same mass – I will have the same number of atoms in it.
PES 1110 Fall 2013, Spendier
Lecture 10/Page 5
Example 3:
Example: A 6860N car is traveling with a speed of 30m/s when the driver removes her
foot from the gas pedal, making the engine force zero. If the frictional force is 350N,
what is the car’s acceleration?
PES 1110 Fall 2013, Spendier
Lecture 10/Page 6
NOTE: If an object is already in motion, just because you push on it, it does not have to
go in the direction you pushed! It will accelerate in the direction you push it.
What about a car pushed from rest?
If I push an object that is initially at rest, it will move to the left, the direction I pushed.
Because the acceleration and velocity are in the same direction.
In the car example above:
If a car is already moving to the right and I push on it to the left, the push is causing an
acceleration to the left, and an acceleration opposite to velocity will slow the object
down. It causes a deceleration.
An object already in motion will not move in the direction of the net force. It will
accelerate in the direction of the net force.
Example 4:
A spaceship is floating sideways from point A to B in the middle of outer space. At B the
spaceship turns on it’s engines for 5 s and moves to point C. At C the engines are again
turned off and the spaceship floats to point D. Which of the following picture correctly
shows the spaceship’s trajectory from A to D?
Solution:
From A to B rocket goes on straight line
since no force ==> uniform motion!
When rockets turned on this causes an acceleration
in y-direction but NOT x-direction ==> acceleration
motion in y, uniform motion in x ==> curved motion
PES 1110 Fall 2013, Spendier
Lecture 10/Page 7
When rockets turned off again we go back to
uniform motion ==> straight line. But now we
have velocity in both x- and y-directions ==>
straight line at an angle From A to B rocket goes
on straight line since no force ==> uniform motion!
Example 5)
Two forces are acting on an object and it is moving with constant velocity. One of the
⌢
forces in F1 = (3.0N) ˆi - (6.1N) j . What is the other force F2 ?
Example 6 a)
A bank robber is trying to push a safe on a frictionless floor. He can apply a horizontal
force of 2.0N. If the safe's mass is 800 kg, what acceleration can he produce on the safe,
if the safe starts from rest?
PES 1110 Fall 2013, Spendier
Lecture 10/Page 8
Example 6 b)
If the bank robber above was sliding the safe on a floor with friction equal to 0.70 N,
what would the acceleration of the safe be?
Example 6 c)
The bank robber above needs to push the safe 4.0m. If the safe is originally at rest, how
long does it take? (The bank robber can accelerate the safe 1.625 x10-3 m/s2 in the
horizontal direction of his push)