PES 1110 Fall 2013, Spendier
Lecture 12/Page 1
Today: Applying Newton’s Laws
There are two main tasks for which we will apply Newton’s Laws
1) To find the magnitude and sometimes direction of unknown forces.
2) To find the acceleration of masses.
It is imperative that we always identify and use the forces acting ON objects (we do not
want to include forces exerted by objects).

- Problems with a  0 are called equilibrium or statics problems (done a lot in Civil &
Mechanical Engineering) - individual forces acting on an object add to zero.

- Problems with a  0 are called dynamic or kinetics problems - individual forces acting
on an object do not add to zero.
Steps for applying Newton’s Laws in problems:
1) Draw a picture.
2) For any object with mass, identify the forces acting on that object. (key step)
3) For each object with mass, draw a free body diagram.

4) Find the net force components from vector addition and set them equal to ma .
5) Solve for unknowns.
Massless Ropes:
We know that ropes are not truly massless but we will ignore the small weight they have
compared to the object which is attached to the rope.
Massless ropes have the very useful property that the tension everywhere must be
equal
What is the rope doing to the block, it is pulling up.
At the same time it is pulling down on the ceiling.
First think of the forces exerted by the rope:

TR on C = force on mass due to bottom of rope

TR on M = force on ceiling due to top of rope
Both will have the same magnitude but pull in opposite direction
PES 1110 Fall 2013, Spendier
Lecture 12/Page 2
Then look at forces on the rope:

TM on R = Tension at the bottom of rope

TC on R = Tension at the top of rope
Newton's second law says what:



 F  ma   F
rope
net



 mrope a  0  TM on R  TC on R
net
Mass of the rope times acceleration of the rope. But because the rope has no mass m=0:


TM on R  TC on R
the two tensions are equal and opposite in direction! These two forces will ALWYAYS
be the same when the rope is massless.

Question: If the rope would not be massless which force would be bigger, TM on R or


TC on R ) It would be TM on R because the top of the rope would have to hold up the object as
well as its own weight.
Example 1:
A skier of mass 65.0 kg is pulled up a slope with a constant speed (parallel to the
ground). The slope of the hill is 26º above the horizontal, and you can ignore friction
a) Draw a clearly labeled free-body diagram for the skier.
b) Calculate the tension in the tow rope.
PES 1110 Fall 2013, Spendier
Lecture 12/Page 3
Example 2:
Two crates one with mass 4.00 kg and the other with mass 6.00 kg, sit on a frictionless
surface connected to a massless rope. A person is pulling horizontally on the 6.00 crate
with force F that gives the crate an acceleration of 2.50 m/s2.
a) What is the acceleration of the 4.00 kg crate?
b) Draw a free-body diagram for the 4.00 kg crate. Use that drawing and Newton's
second law to find the tension T on the rope that connects the two crates.
c) Draw a free-body diagram for the 6.00 kg crate. What is the direction of the net force
on the 6.00 kg crate? Which one is larger in magnitude, force T or force F?
d) Use part c) and Newton's second law to calculate the magnitude of the force F.
PES 1110 Fall 2013, Spendier
Lecture 12/Page 4
Pulleys
What does a pulley do? It pulls. In the case below it changes a horizontal force into a
vertical force. Hence a pulley is:
A machine that changes the direction of applied forces.
For the moment a pulley will always be massless and frictionless. This is the so called
perfect pulley. Perfect pulleys change force direction with no change in magnitude. If the
pulley has mass it will rotate but we do not worry about rotation yet.
Example 3:
How much force must a person exert on the massless rope shown in order to hold the
100-N block at rest?
The force needed to pull down is equal to the tension in the rope.
Hence these two forces have to be the same! At how many places
does this combination of pulleys pull up on the block? It does so
at 4 places. There are 4 upwards pulls on the rope. Each of them
has the same magnitude
mg


 F  ma
net

at rest, a = 0

 100
4T-100  0  T=
 25 N
4
By using a combination of pulleys we can reduce the force needed to lift up a given
object.
PES 1110 Fall 2013, Spendier
Lecture 12/Page 5
Example 4:
Two masses are hung either side of a pulley, connected by a rope. Mass 1 is 4.50 kg and
is accelerating down at 2.60 m/s2.
a) What is mass 2?
b) What is the tension in the rope?