Review Tomorrow
Review Chapters 01 - 07
Chapter 07: Kinetic Energy and Work
Conservation of Energy is one of Nature’s fundamental
laws that is not violated.
Energy can take on different forms in a given system.
This chapter we will discuss work and kinetic energy.
Next chapter we will discuss potential energy.
If we put energy into the system by doing work, this
additional energy has to go somewhere. That is, the
kinetic energy increases or as in next chapter, the
potential energy increases.
The opposite is also true when we take energy out of a system
Power
• The time rate at which work is done by a force
• Average power
Pavg = W/∆t (energy per time)
• Instantaneous power
P = dW/dt = (Fcosφ dx)/dt = F v cosφ= F . v
Unit: watt
1 watt = 1 W = 1 J / s
1 horsepower = 1 hp = 550 ft lb/s = 746 W
• kilowatt-hour is a unit for energy or work:
1 kW h = 3.6 M J
Work done by a variable force
∆Wj = Fj, avg ∆x
W = Σ∆Wj = ΣFj, avg ∆x = lim ΣFj, avg ∆x
∆x →0
xf
W = ∫ F( x )dx
xi
• Three dimensional analysis
xf
yf
zf
xi
yi
zi
W = ∫ Fx dx + ∫ Fy dy + ∫ Fz dz
=∫
rf
ri
v v
F ⋅d r
Problem 07-74
A force directed along the
positive x direction acts on a
particle moving along the xaxis. The force is of the form:
F = a/x2, with a = −9.0 Nm2
Find the work done on the particle by the force as the
particle moves from x=1.0m to x = 3.0m.
Problem: 07-74 Solution
W=∫
xf
xi
v v
xf a
F ⋅ dx = ∫ 2 (cos 0 o )dx
xi x
Problem: 07-74 Applications
Newton’s Law of Gravitation
v
G m1m 2
F=−
r̂
2
r
with G = 6.673 x 10−11 Nm2/kg2
Electrostatic Force between particles (Coulomb’s Law)
v k q1q 2
F = 2 r̂
r
with k = 8.988 x 109 Nm2/C2
Daily Quiz, September 20, 2004
A batter hits a baseball to the outfield. Which graph
could represent its kinetic energy as a function of time?
1
2
6
5
0) none of the above
3
4
7
8
Daily Quiz, September 20, 2004
A batter hits a baseball to the outfield. Which graph
could represent its kinetic energy as a function of time?
1
2
3
6
7
5
0) none of the above
4
8
Daily Quiz, September 20, 2004
A batter hits a baseball to the outfield. Which graph
could represent its kinetic energy as a function of time?
1
K = mv 2
2
where v = v 2x + v 2y =
1
2
(v o cos θ)2 + (v o sin θ − gt )2
3
6
7
5
0) none of the above
4
8
Work done by Spring Force
• Work done by the spring force
xf
xf
xi
xi
Ws = ∫ Fdx = ∫
1 2 1 2
(− kx )dx = kx i − kx f
2
2
• If | xf | > | xi | (further away from x = 0); W < 0
If | xf | < | xi | (closer to x = 0); W > 0
• If xi = 0, xf = x then Ws = − ½ k x2
This work went into potential energy, since the
speeds are zero before and after.
Work done by the applied force: Wa = + ½ k x2
Problem 07-71
Three different masses are hung
from a spring. The resulting
equilibrium positions are
indicated.
A) What is the equilibrium position when no mass is
hung from the spring?
B) What is the weight of the third mass (rightmost, in
green)?
Problem: 07-71 Solution
A) What is the equilibrium position
when no mass is hung from the
spring?
|F1| = |−k∆x1| = k (40−xo) = 110N
|F2| = |−k∆x2| = k (60−xo) = 240N
Subtract the two equations:
240N − 110N = k(60mm − 40mm) => k = 6.5N/mm
Plug k into either equation:
240N = 6.5N/mm(60mm − xo) => xo = 23.0N/mm
Problem: 07-71 Solution
B) What is the weight of the third
mass (rightmost, in green)?
|F| = k|∆x| = (6.5N/mm) |30mm-23mm| = 45.5N
Problem 07-50
A 0.25 kg block is dropped on a relaxed spring that
has a spring constant of k = 250.0 N/m (2.5 N/cm).
The block becomes attached to the spring and
compresses it 0.12 m before momentarily stopping.
While being compressed,
A) What is the work done on it by gravity?
B) What is the work done on it by the spring force?
C) What is the speed of the block just prior to hitting the spring?
Problem 07-50
A 0.25 kg block is dropped on a relaxed spring that
has a spring constant of k = 250.0 N/m (2.5 N/cm).
The block becomes attached to the spring and
compresses it 0.12 m before momentarily stopping.
While being compressed,
A) What is the work done on it by gravity?
B) What is the work done on it by the spring force?
Problem 07-50
A 0.25 kg block is dropped on a relaxed spring that
has a spring constant of k = 250.0 N/m (2.5 N/cm).
The block becomes attached to the spring and
compresses it 0.12 m before momentarily stopping.
C) What is the speed of the block just prior to hitting the spring?
Problem 07-60
v
Force F = (3.0î + 7.0 ĵ + 7.0k̂ )
Initial Displacement
v
d i = (3.0î − 2.0 ĵ + 5.0k̂ )
in ∆t = 4.0s
Final Displacement
v
d f = (−5.0î + 4.0 ĵ + 7.0k̂ )
A) Work done on the particle
B) Average power
C Angle between di and df
Problem 07-60
v
Force F = (3.0î + 7.0 ĵ + 7.0k̂ )
Initial Displacement
v
d i = (3.0î − 2.0 ĵ + 5.0k̂ )
in ∆t = 4.0s
Final Displacement
v
d f = (−5.0î + 4.0 ĵ + 7.0k̂ )
Problem 07-70
m = 230kg, L = 12.0m
variable horizontal force F
horizontal distance d = 4.0m
A) magnitude of F in final position
B) Total Work done on crate
C) Gravitational Work
D) Work done by rope
E) Work done by force F
Problem 07-70
m = 230kg, L = 12.0m
variable horizontal force F
horizontal distance d = 4.0m
Problem 07-70
m = 230kg, L = 12.0m
variable horizontal force F
horizontal distance d = 4.0m