Transcript L14_power

Work, Energy, and Power
§ 6.2–6.4
Kinetic Energy
Energy of a moving mass
§ 6.2
CPS Question
To accelerate an object from 10 to 20 m/s
requires
A. more work than to accelerate from 0 to
10 m/s.
B. the same amount of work as to
accelerate from 0 to 10 m/s.
C. less work than to accelerate from 0 to 10
m/s.
Work of Acceleration
• To accelerate to speed v with constant force F
slope = a =
v
F
=
m
t
v
area = Dd
speed
time
• Work = F·Dd
t
mv
1
Dd = 2 vt
F = m (slope) = t
mv
1
1
2
mv
vt
=
• Work = 2
2
t
Another Perspective
• So, for the 0–10 vs. 10–20 m/s case:
• If same force, then same time
– a’s and Dv’s are equal, so Dt’s are equal
• Average speeds are 5 vs. 15 m/s
• The 10–20 m/s case travels 3 as far
A Moving Object Can Do Work
Source: Griffith, The Physics of Everyday Phenomena
Kinetic Energy
the work to bring a motionless object to
speed
K=
1
2
mv2
equivalent to
the work a moving object does in stopping
CPS Question
Which has more kinetic energy?
A.
10 kg
10 m/s
5 kg
10 m/s
5 kg
40 m/s
10 kg
20 m/s
B.
C.
D.
Rebounding Ball
What is the sign of the work done on the ball
by the wall as it slows (squishes)?
A.
B.
C.
D.
Positive (W > 0).
Negative (W < 0).
Zero (W = 0).
Can’t tell (W = ?).
Rebounding Ball
What is the sign of the work done on the ball
by the wall as it rebounds (expands)?
A.
B.
C.
D.
Positive (W > 0).
Negative (W < 0).
Zero (W = 0).
Can’t tell (W = ?).
Happy/Sad Balls
Which ball has the greatest change in kinetic
energy DK during impact?
A. The happy (rebounding) ball.
B. The sad (dead) ball.
C. Both had the same DK.
Happy/Sad Balls
Which ball has the most (largest absolute
value) work done on it during impact?
A. The happy (rebounding) ball.
B. The sad (dead) ball.
C. Both had the same W.
Worksheet Problem 1
A luge and its rider, total mass 90 kg, emerges
onto a level track with v0 = 36 m/s. It undergoes a
constant deceleration of 2.0 m/s2 until it stops.
a) What is the magnitude of the force acting on it?
b) What distance does it travel while decelerating?
i. First find the general kinematic formula for distance Dx
traveled in stopping from speed v0 at acceleration a.
ii. Then find Dx in this case.
c) What work does the force do?
Worksheet Problem 1
A luge and its rider, total mass 90 kg, emerges
onto a level track with v0 = 36 m/s. It undergoes a
constant deceleration of 4.0 m/s2 until it stops.
d) What is the magnitude of the force acting on it?
e) What distance does it travel while decelerating?
–
Use the general formula found earlier.
f) What work does the force do?
g) What was the initial kinetic energy K0 of the luge?
CPS Question
The piglet has a choice of three frictionless
slides to descend. Along which slide would
the piglet finish with the highest speed?
A
B
C
D. The final speed is the same for all.
Worksheet Problem 2
A piglet slides down a frictionless ramp of height H
and angle a above the horizontal. What is its
speed vf at the end?
a) Find the general kinematic equation for vf when
accelerating from rest at acceleration a through a
distance Dx.
b) Geometrically find a and Dx in terms of H and a.
c) Find vf in this case.
d) What is the dependency of vf on a?
CPS Question
The piglet has a choice of three frictionless
slides to descend. Along which slide would
the piglet finish soonest?
A
B
C
D. The time is the same for all.
CPS Question
Now the piglet/slide interface has a little
friction. Along which slide would the piglet
finish with the highest speed?
A
B
C
D. The final speed is the same for all.
Work-Energy Theorem
• If an amount of work W is done on an
otherwise isolated system, the system’s
energy changes by an amount DE = W.
• The net work done on an object equals its
change in kinetic energy Wtot = DK.
Work in General
curving paths, changing forces
§ 6.3
What’s the point?
• What is work when force is not constant or
the path is not straight?
Work in General
• For constant force, W = F·s.
• F may vary with position or time.
• Path may not be straight.
• In general, dW = F·ds.
• So, W = F·ds.
• (Sum of work done over each interval.)
Elastic Force
Stretching and squishing
§ 6.3
Structure of Solids
• Atoms and molecules connected by
chemical bonds
• Considerable force needed to deform
compression
tension
Elasticity of Solids
Small deformations are proportional to force
small stretch
larger stretch
Hooke’s Law: ut tensio, sic vis (as the pull,
so the stretch)
Robert Hooke, 1635–1703
Hooke’s Law Formula
F = –kx
F = force exerted by the spring
k = spring constant; units: N/m; k > 0
x = displacement from equilibrium position
negative sign: force opposes distortion
CPS Question
backward
forward
What direction of
forward
force is needed to Spring’s
hold the object
Force
backward
(against the
spring) at its
Displacement
plotted
displacement?
A. Forward (right).
C. No force (zero).
B. Backward (left).
D. Can’t tell.
Work to Deform a Spring
• Push or pull a distance x from equilibrium
slope = k
kx
area = w
force
displacement
• Work =
1
2
• Work =
1
2 kx·x
F·x ; F = kx
1
=
2
kx2
x
CPS Question
A spring with force constant k is stretched
from x = 0 to x = D. What is the work done
by the spring as it stretches?
A.
B.
C.
D.
E.
1/2 kD2.
–1/2 kD2.
0.
It cannot be determined.
None of these.
CPS Question
Two springs, one with a spring constant k1 and the
other with a spring constant k2 = 2 k1, are slowly
stretched to the same final tension. Which spring
has more work done on it?
A. The stiffer spring (k = 2 k1)
B. The softer spring (k = k1)
C. The same work was done on both.
Worksheet Problem 3
A 1.53-kg block is released from rest just atop a
relaxed spring with k = 2.50 N/cm. The block
compresses the spring 12.0 cm before
momentarily stopping.
a) While the spring is compressing, what work is done on
the block:
i. by gravity?
ii. by the spring?
b) What force does the spring finally exert?
c) What is the block’s final acceleration?
d) What is the block’s initial acceleration?
Centripetal Force
work of acceleration
§ 6.3
Group CPS Question
A toy of mass m moving at constant speed v in a
circle of radius r has a constant magnitude of
centripetal acceleration of v2/r. Its velocity
reverses every half-cycle.
How much work does the centripetal force do on
the toy every half-cycle?
A.
B.
C.
D.
mv2.
–mv2.
pmv2.
None of these.
Kinetic Energy and Direction
• K Depends on speed
• Direction of velocity is irrelevant
• Changing direction only requires force, but
no work.
• 1/2 mv2 = 1/2 mvv is a scalar
Net Force and Net Work
• Net force is nonzero if a body accelerates
• net work is nonzero if a body changes
speed
• The net force must overlap with the
displacement to do work!
Worksheet Problem 4
Your cousin Throckmorton, m = 20 kg, plays on a
R = 1.5-m swing. What is the net work done on
him as he swings from an angle q = p/6 from
vertical down to q = 0?
a)
b)
c)
d)
What is the net force as a function of q?
How far (total path length) does he travel?
Set up the integral for the total work done on him.
Evaluate the total work.
Power
how quickly work is done
§ 6.4
Power
Rate of doing work
Power =
DE
Dt
w
=
Dt
DE = change in energy ( = work)
Dt = time interval
Units of Power
P=
DE
Dt
Energy
= J/s = W = watt
time
kg m2
kg m2
W= 2
=
s s
s3
Power
A different but equivalent formula
W
F·Ds
P=
=
= F·v
Dt
Dt
F = force
Ds = displacement
v = velocity
CPS Question
Dragging a box across a level floor against
friction at 1 m/s requires a power of 20 W.
How much power is required to drag the
same box at 2 m/s?
A. 10 W.
B. 20 W.
C. 40 W.
D. 80 W.