Newton’s Laws (cont…)
• Blocks, ramps, pulleys and other problems
Physics 1D03 - Lecture 8
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Example
Two blocks connected by a rope are being pulled by a horizontal
force FA. Given that FA=60 N, m1=12kg and m2=18kg, and that
μk=0.1, find the tension in the rope between them and the
acceleration of the system.
T
m1
m2
FA
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Solution
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Quiz
Two blocks of mass m1 and m2 (m1>m2) are being pushed by a
horizontal force FA over a frictionless surface.
Consider the contact force between m1 and m2.
FA
m1
F12
m2
F21
A) F12 > F21
B) F12 = -F21
C) F12 > F21
D) F12 = F21
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Example
Two blocks of mass m1 and m2 (m1>m2) are being pushed by a
horizontal force FA over a frictionless surface. Determine:
a) the magnitude of the acceleration of the system
b) the magnitude of the contact force between the blocks
FA
m1
m2
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Solution
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Example
A block of mass m=5kg is being pulled by two forces. F1 = 6N
which acts along the positive x-axis, and F2 = 15N which acts at
an angle of 135o to the positive x-axis.
What is the acceleration of the block ?
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Example
A block of mass m=2kg is suspended by two ropes as
shown below. What are the tensions in each rope ?
30o
60o
m
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Elevator go up, elevator go down
• A person of mass 70kg is standing on a scale in an
elevator at rest. What is her weight ?
• What is her weight when the elevator is accelerating
up at 5m/s2 ???
• What is her weight when the elevator is accelerating
down at 5m/s2 ???
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10 min rest
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Pulleys
• To solve pulley problems, we assume that:
1) the pulley is frictionless
2) the pulley is massless
• Hence, the force of tension on both sides of the
pulley is the same
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Example
• Find the acceleration of a system of two masses
m=5kg and M=10kg. The angle θ=25o. No friction!
• Also, find the tension, T, in the string.
M
m
q
There are two ways of solving the problem – find T first,
or last.
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Solution
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Quiz
Two masses m and M are connected as shown. The
angle is θ. If there is friction on the incline, can we
write the equation of motion without knowing the
masses or the angle?
M
m
q
A) Yes, we can
B) No, we can’t
C) What’s an equation of motion ???
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Example: A block of mass m1 on a rough horizontal surface is pulled
with a force FA at an angle θ to the horizontal. A ball of mass m2 is
connected to the other side, hanging over a lightweight frictionless
pulley. The coefficient of friction is given by μk. Assume m1 will
move to the right. Determine the acceleration of the system.
FA
θ
m1
m2
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Solution
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Atwood’s Machine
Calculate the acceleration of the blocks.
Assume :
- no friction
- massless rope and pulley
- rope doesn’t stretch
Plan: • free-body diagram for each mass
• relate tensions, accelerations
• use Newton’s second Law
m1
Physics 1D03 - Lecture 8
m2
17
Forces on m1
T1
Forces on m2
T2
a2
a1
m2g
m1g
•
•
Tensions are equal (“ideal” pulley, light rope)
Accelerations are equal in magnitude (why?), opposite in
direction
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T
T
a
a
m2g
m1g
m1g  T  m1a
T  m2g  m2a
.
.
.
 m1  m2 
g
Eliminate T to get  a  
 m1  m2 

a is proportional to g, but can be small (and easy to measure)
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10 min rest
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Motion in 2D
• Constant acceleration in 2-D
• Free fall in 2-D
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
The Position vector r
points from the origin to
the particle.
y
yj
path
(x,y)

r
xi
x

The components
of r are the coordinates (x,y) of the

particle: r  x i  y j

For a moving particle, r (t ), x(t), y(t) are functions of
time.
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Components: Each vector relation implies 2
separate relations for the 2 Cartesian components.

r  xi y j
(i, j are unit vectors)
We get velocity components by differentiation:

 dr
v
dt
 dx   dy 
  i  j
 dt   dt 
 vx i  v y j
the unit vectors are
constants
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Constant Acceleration + Projectile Motion

If a is constant (magnitude and direction), then:

 
v (t )  vo  a t


 2
1
r (t )  vo t  2 a t
Where

vo is the initial value at t = 0.
In 2-D, each vector equation is equivalent to a pair of
component equations:
x(t )  vox t 
1
2
ax t
2
y (t )  voy t  1 2 a y t 2
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Quiz
A stone is thrown from the top of a building with a θ
angle above the horizontal. In what direction (if any)
does the stone experience an acceleration?
A) Horizontal direction only
B) Vertical direction only
C) Both, horizontal and vertical directions
D) No acceleration once it leaves your hand
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Projectile Motion: The Equations

If a is constant (magnitude and direction), then:

 
v (t )  vo  a t


 2
1
r (t )  vo t  2 a t
In 2-D free-fall type questions, we separate the motion
into horizontal and vertical parts:
v x  vox
x  vox t
< Horizontal, a = 0
and
v y  voy  gt
y  voy t  1 2 g t 2
< Vertical, a = -g
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Quiz
Tarzan has a new slingshot. George the gorilla hangs from
a tree, and bets that Tarzan can’t hit him. Tarzan aims at
George, and is sorry that he didn’t pay more attention in
physics class. Why ? (neglect air resistance)
A) The stone will go over the gorilla’s head
B) The stone will go below the gorilla
C) The earth’s rotation must be taken into account
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Projectile Motion
y=voyt-½gt2
x=voxt
y
vo
x
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Example
A stone is thrown upwards from the top of a 45.0 m high
building with a 30º angle above the horizontal. If the
initial velocity of the stone is 20.0 m/s, how long is the
stone in the air, and how far from the base of the
building does it land ?
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Height, Range, Time
Determine the maximum height of a projectile
Determine how far it goes (range)
Determine the total time in flight
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Problem
A stone is thrown from the top of a building with a θ
angle above the horizontal with an initial velocity vo.
How high will it go?
How far will it go?
How long does is it in the air for?
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