Download In vertical circular motion the gravitational force must also be

Vertical Circular Motion
In vertical circular motion the gravitational
force must also be considered.
An example of vertical circular motion is the
vertical “loop-the-loop” motorcycle stunt.
Normally, the motorcycle speed will vary
around the loop.
The normal force, FN, and the weight of
the cycle and rider, mg, are shown at four
locations around the loop.
Vertical Circular Motion
1)
2)
3)
4)
2
1
v
FN 1 − mg = m
r
FN 2
2
2
v
=m
r
2
3
v
FN 3 + mg = m
r
FN 4
2
4
v
=m
r
Vertical Circular Motion
There is a minimum speed the rider
must have at point 3 in order to stay on
the loop.
This speed may be found by setting
FN3 = 0 in the centripetal force equation
for point 3, i.e. in FN3 + mg = mv32/r
è  v3min = (rg)1/2
e.g. for a track with r = 10 m,
v3min = ((10)(9.8))1/2 = 9.9 m/s = 22 mph
Example: Find the orbital periods of Mercury (~0.4 AU from the Sun) and
Neptune (~30 AU from the Sun) and compare with that of the Earth.
M = Sun mass, m = planet mass, r = distance to Sun, v = planet speed in orbit
mv 2
mM
v2
M
FC =
=G 2 ⇒ 2 =G 3
r
r
r
r
2π r
τ = orbital period =
v
r3
∴ τ = 2π
GM
1 AU = distance from Earth to Sun =1.5 x 1011 m
11 3
τ Earth = 2π
(1.5 ×10 )
= 3.2 ×10
(6.67 ×10 ) (1.99 ×10 )
7
30
−11
s
11 3
τ Mercury = 2π
τ Neptune = 2π
(0.4 ×1.5 ×10 )
= 8.0 ×10
(6.67 ×10 ) (1.99 ×10 )
30
−11
(30 ×1.5 ×1011 )
s = 0.25τ Earth
3
(6.67 ×10 ) (1.99 ×10 )
−11
6
30
= 5.2 ×10 9 s = 160τ Earth
End of material for
Midterm 1
Chapter 6
Work and Energy
Work Done by a Constant Force
Work involves force and displacement.
W = Fs
1 N ⋅ m = 1 joule (J )
Work Done by a Constant Force
Work Done by a Constant Force
More general definition of the work done on an object, W, by a
constant force, F, through a displacement, s, with an angle, θ,
between F and s:
W = ( F cosθ ) s
cos 0 = 1
cos 90 = 0
cos180 = −1
Note that W is a scalar!
Work Done by a Constant Force
Example: Pulling a Suitcase-on-Wheels
Find the work done if the force is 45.0 N, the angle is 50.0
degrees, and the displacement is 75.0 m.
W = ( F cosθ ) s = !"( 45.0 N ) cos50.0 #$( 75.0 m )
= 2170 J
Work Done by a Constant Force
A weightlifter doing positive and
negative work on the barbell:
Lifting phase --> positive work.
W = ( F cos0 ) s = Fs
Lowering phase --> negative work.
W = ( F cos180 ) s = −Fs
Example. Work done skateboarding. A skateboarder is coasting down a
ramp and there are three forces acting on her: her weight W (675 N
magnitude) , a frictional force f (125 N magnitude) that opposes her motion,
and a normal force FN (612 N magnitude). Determine the net work done by
the three forces when she coasts for a distance of 9.2 m.
W = (F cos θ)s , work done by
each force
W = (675 cos 65o)(9.2)
= 2620 J
W = (125 cos 180o)(9.2)
= - 1150 J
W = (612 cos 90o)(9.2)
=0J
The net work done by the three forces is: 2620 J - 1150 J +0 J = 1470 J
Work Done by a Constant Force
Example: Accelerating a Crate
The truck is accelerating at
a rate of +1.50 m/s2. The mass
of the crate is 120 kg and it
does not slip. The magnitude of
the displacement is 65 m.
What is the total work done on
the crate by all of the forces
acting on it?
Work Done by a Constant Force
The angle between the displacement
and the normal force is 90 degrees.
The angle between the displacement
and the weight is also 90 degrees.
Thus, for FN and W,
W = ( F cos90 ) s = 0
Work Done by a Constant Force
The angle between the displacement
and the friction force is 0 degrees.
fs = ma = (120 kg) (1.5m s2 ) = 180N
W = !"(180N ) cos0#$( 65 m ) = 1.2 ×10 4 J
The Work-Energy Theorem and Kinetic Energy
Consider a constant net external force acting on an object.
The object is displaced a distance s, in the same direction as
the net force.
∑F
s
The work is simply
W = (Σ F)s = (ma)s
The Work-Energy Theorem and Kinetic Energy
(
2
f
2
o
)
2
f
2
o
W = m(as ) = m v − v = mv − mv
1
2
1
2
1
2
v 2f = vo2 + 2 ( as )
constant acceleration
kinematics formula
2
2
1
as
=
v
−
v
( ) 2( f o)
DEFINITION OF KINETIC ENERGY
The kinetic energy KE of and object with mass m
and speed v is given by
KE = mv
1
2
2
SI unit: joule (J)
The Work-Energy Theorem and Kinetic Energy
THE WORK-ENERGY THEOREM
When a net external force does work on an object, the kinetic
energy of the object changes according to
2
f
2
o
W = KE f − KE o = mv − mv
1
2
1
2
(also valid for curved paths and non-constant accelerations)
The Work-Energy Theorem and Kinetic Energy
Example: Deep Space 1
The mass of the space probe is 474 kg and its initial velocity
is 275 m/s. If the 56.0 mN force acts on the probe through a
displacement of 2.42×109m, what is its final speed?
The Work-Energy Theorem and Kinetic Energy
2
f
2
o
W = mv − mv
1
2
W = [(Σ F)cos θ]s
1
2
The Work-Energy Theorem and Kinetic Energy
[(∑F)cos
F)cosθθ
[(Σ
]s]s =

1
2
mvf2 − 12 mvo2
(5.60 ×10 N) cos0 (2.42 ×10 m) = ( 474 kg) v
-2
9
1
2
2
f
−
1
2
( 474 kg) (275m s)
v f = 805 m s
2
The Work-Energy Theorem and Kinetic Energy
Conceptual Example: Work and Kinetic Energy
A satellite is moving about the earth
in a circular orbit and an elliptical orbit.
For these two orbits, determine whether
the kinetic energy of the satellite
changes during the motion.
Gravitational Potential Energy
Find the work done by gravity in
accelerating the basketball
downward from ho to hf.
W = ( F cosθ ) s
Wgravity = mg ( ho − h f )
Gravitational Potential Energy
Wgravity = mg ( ho − h f )
We get the same
result for Wgravity
for any path taken
between ho and hf.
Gravitational Potential Energy
Example: A Gymnast on a Trampoline
The gymnast leaves the trampoline at an initial height of 1.20 m
and reaches a maximum height of 4.80 m before falling back
down. What was the initial speed of the gymnast?
Gravitational Potential Energy
W = 12 mvf2 − 12 mvo2
mg (ho − h f )= − 12 mvo2
Wgravity = mg (ho − h f )
vo = −2g ( ho − h f )
vo = −2 ( 9.80 m s2 ) (1.20 m − 4.80 m ) = 8.40 m s
Gravitational Potential Energy
Wgravity = mgho − mgh f
DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY
The gravitational potential energy PE is the energy that an
object of mass m has by virtue of its position relative to the
surface of the earth. That position is measured by the height
h of the object relative to an arbitrary zero level:
PE = mgh
1 N ⋅ m = 1 joule (J )
The Ideal Spring
HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING
The restoring force on an ideal spring is
SI unit for k: N/m
Fx = − k x
The Ideal Spring
Example: A Tire Pressure Gauge
The spring constant of the spring
is 320 N/m and the bar indicator
extends 2.0 cm. What force does the
air in the tire apply to the spring?
The Ideal Spring
Applied
x
F
= kx
= (320 N m )(0.020 m ) = 6.4 N
Potential energy of an ideal spring
A compressed spring can do work.
Potential energy of an ideal spring
Welastic = ( F cosθ ) s = 12 ( kxo + kx f ) cos0 ( xo − x f )
|F| = kx
F0
Welastic = 12 kxo2 − 12 kx 2f
Ff
xf
x0
F = 12 ( F0 + Ff )
s
F
Potential energy of an ideal spring
DEFINITION OF ELASTIC POTENTIAL ENERGY
The elastic potential energy is the energy that a spring
has by virtue of being stretched or compressed. For an
ideal spring, the elastic potential energy is
PE elastic = 12 kx 2
SI Unit of Elastic Potential Energy: joule (J)