Download F - Physics

Midterm 1
Physics 1200 – Humanic
XXXX lecture
February 25, 2015
Name _________________________________Rec. Instructor_____________________
* Exam is closed book and closed notes. A calculator is allowed.
* Write your name and the name of your recitation instructor on every page of the exam.
* This exam consists of two sections:
Section I – 8 multiple choice questions (40 points)
Section II – 2 problems where you are required to show work (50 points)
* You have 55 minutes to complete both sections.
Σ Fx = max
Σ Fy = may F = Gm1m2/r2 G = 6.67 x 10-11 Nm2/kg2 g(Earth) = 9.80 m/s2
vx = v0x + axt
x = ½ (v0x + vx)t
x = v0xt + ½ axt2
vx2 = v0x2 + 2axx
vy = v0y + ayt
y = ½ (v0y + vy)t
y = v0yt + ½ ayt2
vy2 = v0y2 + 2ayy
f k = µk FN
_
v = Δr/Δt
W = F(cos θ)s
h2 = ho2 + ha2
_
a = Δv/Δt
W = ΔKE
sin θ = ho/h
F = mg + ma
KE = ½mv2
cos θ = ha/h
ac = v2/r
PE = mgh
tan θ = ho/ha
Fc = mv2/r
F = -kx
fsMAX = µsFN
PEelastic = ½kx2
Chapter 6
Work and Energy
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
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)
Conservative Versus Nonconservative Forces
DEFINITION OF A CONSERVATIVE FORCE
Version 1 A force is conservative when the work it does
on a moving object is independent of the path between the
object’s initial and final positions.
Version 2 A force is conservative when it does no work
on an object moving around a closed path, starting and
finishing at the same point.
Conservative Versus Nonconservative Forces
Conservative Versus Nonconservative Forces
Version 1 A force is conservative when the work it does
on a moving object is independent of the path between the
object’s initial and final positions.
Wgravity = mg ( ho − h f )
Conservative Versus Nonconservative Forces
Version 2 A force is conservative when it does no work
on an object moving around a closed path, starting and
finishing at the same point.
Wgravity = mg ( ho − h f )
ho = h f è Wgravity = 0
Conservative Versus Nonconservative Forces
An example of a nonconservative force is the kinetic
frictional force.
W = ( F cosθ ) s = fk cos180 s = − fk s
The work done by the kinetic frictional force is always negative.
Thus, it is impossible for the work it does on an object that
moves around a closed path to be zero.
The concept of potential energy is not defined for a
nonconservative force.
Conservative Versus Nonconservative Forces
In normal situations both conservative and nonconservative
forces act simultaneously on an object, so the work done by
the net external force can be written as
W = Wc + Wnc
W = KE f − KE o = ΔKE
Wc = Wgravity = mgho − mgh f = PE o − PE f = −ΔPE
Conservative Versus Nonconservative Forces
W = Wc + Wnc
ΔKE = −ΔPE + Wnc
A different version of the WORK-ENERGY THEOREM
Wnc = ΔKE + ΔPE
The Conservation of Mechanical Energy
Wnc = ΔKE + ΔPE = ( KE f − KE o ) + ( PE f − PE o )
Wnc = ( KE f + PE f ) +_ ( KE o + PE o )
Wnc = E f − E o = ΔE
E = KE + PE è the total mechanical energy
If the net work on an object by nonconservative forces
is zero, then its energy does not change:
Ef = Eo
The Conservation of Mechanical Energy
THE PRINCIPLE OF CONSERVATION OF
MECHANICAL ENERGY
The total mechanical energy (E = KE + PE) of an object
remains constant as the object moves, provided that the net
work done by external nonconservative forces is zero.
The Conservation of Mechanical Energy
The Conservation of Mechanical Energy
Example: A Daredevil Motorcyclist
A motorcyclist is trying to leap across the canyon by driving
horizontally off a cliff 38.0 m/s. Ignoring air resistance, find
the speed with which the cycle strikes the ground on the other
side.
The Conservation of Mechanical Energy
Ef = Eo
2
f
2
o
mgh f + mv = mgho + mv
1
2
1
2
gh f + 12 v 2f = gho + 12 vo2
The Conservation of Mechanical Energy
gh f + 12 v 2f = gho + 12 vo2
v f = 2g ( ho − h f ) + vo2
v f = 2 ( 9.8m s
2
) (35.0m) + (38.0 m s)
2
= 46.2 m s
The Conservation of Mechanical Energy
Conceptual Example: The Favorite Swimming Hole
The person starts from rest, with the rope
held in the horizontal position,
swings downward, and then lets
go of the rope. Three forces
act on him: his weight, the
tension in the rope, and the
force of air resistance.
Can the principle of
conservation of energy
be used to calculate his
final speed?
The Conservation of Mechanical Energy
Example: Vertical spring with a mass
A 0.20-kg ball is attached to a vertical spring. The spring constant
is 28 N/m. When released from rest, how far does the ball fall
before being brought to a momentary stop by the spring?
The Conservation of Mechanical Energy
E f = E0
1
2
mv 2f + mgh f + 12 k(Δy f )2 = 12 mv02 + mgh0 + 12 k(Δy0 )2
v0 = v f = 0
Δy0 = 0 , since spring initially unstretched
Δy f = h0
hf = 0
∴ 12 kh02 = mgh0
2mg
h0 =
k
2 ( 0.20 kg) ( 9.8m s2 )
=
= 0.14 m
28N m
Nonconservative Forces and the Work-Energy Theorem
Example of a nonconservative force problem: Fireworks
A 0.20 kg rocket in a fireworks display is launched from rest
and follows an erratic flight path to reach the point P, as in the
figure. P is 29 m above the starting point. In the process, 425 J
of work is done on the rocket by the nonconservative force
generated by the burning propellant. Ignoring air
resistance and the mass loss due to the burning
propellant, find the speed vf of the rocket at point P.
THE WORK-ENERGY THEOREM
Wnc = E f − E o
Wnc = ( mgh f + 12 mv 2f ) −
(mgh
o
2
o
+ mv
1
2
)
Nonconservative Forces and the Work-Energy Theorem
2
f
2
o
Wnc = mgh f − mgho + mv − mv
1
2
Wnc = mg ( h f − ho ) + mv
1
2
1
2
2
f
425 J = ( 0.20 kg) ( 9.80 m s2 ) ( 29.0 m )
+ 12 ( 0.20 kg) v 2f
v f = 61 m s
Power
DEFINITION OF AVERAGE POWER
Average power is the rate at which work is done, and it
is obtained by dividing the work by the time required to
perform the work.
Work W
P=
=
Time
t
joule s = watt (W)
Power
From the work-energy theorem, doing work results in a
change in energy for the system, so alternatively:
Change in energy
P=
Time
1 horsepower = 550 foot ⋅ pounds second = 745.7 watts
Another way to express the power generated by a force
acting on an object moving at some average speed,
P = Fv