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Chapter 3 Mechanical Objects, Part 1
October 2: Spring Scales – Hooke’s law
1
Question:
What is exactly a spring scale measuring?
Discussion:
Measuring mass and measuring weight.
• An object’s mass is the same everywhere.
• An object’s weight varies with gravity.
2
Equilibrium states
• An object is in an equilibrium state
when it experiences zero net force.
• At equilibrium, an object can be
either at rest, or coasting.
• A spring scale measures the force it
receives. It measures weight using
equilibrium.
• A spring scale is accurate only when
everything is in equilibrium.
3
Question
You are standing on a bathroom scale in an elevator.
When the elevator starts moving upward, the scale will
read
A) Exactly your weight.
B) More than your weight.
C) Less than your weight.
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Springs:
• A free spring has an equilibrium length, when its ends
are not pulled or pushed.
• When distorted, the ends of the spring experience
forces that tend to restore the spring to its equilibrium
length. These forces are called restoring forces.
restoring
force
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Hooke’s law (the law of elasticity)
The restoring force exerted by a spring is
proportional to how far it has been distorted
from its equilibrium length. The restoring force
is directed to oppose the distortion.
restoring force   spring constant  distortion
F  k  x
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Robert Hooke (1635-1703) English natural philosopher,
architect and polymath. Discovered “the law of elasticity”.
Discovered cell. No portrait exists.
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Question:
How much will the spring stretch if I add more bricks?
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More examples of Hooke’s law
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Read: Ch3: 1
Homework: Ch3: E5,7;P3
Due: October 9
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October 5: Ball Sports: Bouncing –
Coefficient of restitution
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Springs: Elastic potential energy
I stretched a spring for a distance of x. The spring has a spring
constant of k.
Question 1:
Did I do work on the spring?
Question 2:
How much work have I done on the spring?
Question 3:
Where has my work gone?
x
Elastic potential energy
1
 k  x2
2
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Energy change in a bouncing ball
Collision energy: The kinetic energy absorbed during the collision.
Rebound energy: The kinetic energy released during the rebound.
When a ball strikes a rigid wall, the ball’s
• kinetic energy decreases by the collision
energy.
• elastic potential energy increases as it dents.
When the ball rebounds from the wall, the ball’s
• elastic potential energy decreases as it
undents.
• kinetic energy increases by the rebound
energy.
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Question: Why can’t a ball that’s dropped on a hard
floor rebound to its starting height?
Answer:
Rebound energy < Collision energy
because of loss of the energy into thermal energy.
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Coefficient of restitution: Measuring a ball’s liveliness
Coefficient of restitution
• Is a conventional measure of a ball’s liveliness.
• Is the ratio between the outgoing and the incoming
speeds:
rebound speed of the ball
coefficien t of restitutio n 
collision speed of the ball
• Is measured in bouncing from a rigid surface.
• The rebound speed is then
rebound speed  collision speed  coefficien t of restitutio n
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rebound energy
energy ratio 
 (speed ratio) 2
collision energy
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Question 1:
A basket ball hits a rigid floor at a velocity of 2 m/s.
What is its rebound velocity if the coefficient of
restitution is 0.80?
Question 2:
A ball’s coefficient of restitution is 0.5. It is dropped
from 1 meter high onto a rigid floor. How high will it
bounce? (Hint: energy ratio = (speed ratio)2.)
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Read: Ch3: 2
Homework: Ch3: E11;P4
Due: October 14
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October 7: Ball Sports: Bouncing – Effects from
surfaces
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Review questions:
1. A tennis ball has a coefficient of restitution of 0.75.
It hits a rigid floor at a speed of 2 m/s. How much is
its rebound speed?
2. If the tennis ball is dropped from 1 meter high onto
the floor, how high will it bounce?
(Hint: energy ratio = (speed ratio)2.)
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Ball bouncing from an elastic surface
•
•
•
•
Both the ball and the surface dent during the collision.
Work done in distorting each object is proportional to the
dent distance. Whichever object dents more receives more
collision energy.
Both the denting ball and the denting surface store and
return energy.
A soft, lively surface can help the ball to bounce.
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Examples of lively surfaces
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Ball bouncing from a moving surface
•
•
•
Incoming speed → relative approaching speed
Outgoing speed → relative separating speed
The coefficient of restitution now becomes
separating speed
coefficien t of restitutio n 
approachin g speed
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Relative velocities
Two cars are traveling at 60 mph and 50 mph,
respectively, according to a pedestrian.
1) When they collide head-on, how much is their
approaching speed?
2) When they collide head-on-tail, how much is their
approaching speed?
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Ball bouncing from a moving surface: Example
•
The approaching speed is 200 km/h.
•
Baseball’s coefficient of restitution is 0.55. The
separating speed is 110 km/h.
•
The bat heads toward the pitcher at 100 km/h. The ball
heads toward the pitcher at 210 km/h.
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The ball’s effects on the bat
•
The ball 1) pushes the bat back and
2) rotates the bat.
•
When the ball hits the bat’s center of
percussion, the bat’s backward and
rotational motions balance, so that
the bat’s handle doesn’t jerk.
•
When the ball hits the bat’s
vibrational node, the bat doesn’t
vibrate.
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Read: Ch3: 2
Homework: Ch3: E14,19
Due: October 14
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October 9: Carousels and Roller Coasters –
Circular motion
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Examples of circular motions
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Uniform circular motions
• An object is in a uniform circular
motion if its trajectory is circular and
its speed is a constant.
• When an object is in a uniform circular
motion, it has a net acceleration toward
the center of the circle, which is called
the centripetal acceleration.
• The centripetal acceleration is caused
by a centripetal force, which is the net
force exerted on the object.
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Centripetal acceleration and centripetal force
• The centripetal acceleration is given by
speed 2
Centripeta l accelerati on 
 angular speed 2  radius
radius
v2
a
 2 r
r
• The centripetal force is given by
speed 2
Centripeta l force  mass 
 mass  angular speed 2  radius
radius
v2
F  m   m  2  r
r
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More about centripetal force
• Centripetal force is needed to keep the circular motion of
an object, otherwise the object will move on a straight
line according to Newton’s first law of motion.
• Centripetal force is not a new kind of force, it is rather a
net sum of force provided by whatever traditional forces
we have known.
• There is no such force called centrifugal force exerted on
the object. Centrifugal force is only related to our
feeling.
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Question:
You are running on a circular track with a radius of 20
m. Your mass is 70 kg. Your speed is 2 m/s.
1) How much is your acceleration?
v 2 (2 m/s) 2
Centripeta l accelerati on  
 0.2 m/s 2
r
20 m
2) How much centripetal force is needed?
v2
Centripeta l force  m  70 kg  0.2 m/s 2  14 N
r
3) Who exerts this centripetal force on you?
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Questions:
A child with a mass of 30 kg is riding on a playground
carousel with a radius of 1.5 m. The carousel turns one
circle every two second.
Question 1:
How much is the speed of the child?
Question 2:
How much is the acceleration of the child?
Question 3:
How much is the centripetal force on the child?
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Examples of centripetal forces
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Read: Ch3: 3
Homework: Ch3: E31;P6
Due: October 14
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