PHY 53
Summer 2010
Assignment 5
Reading: Energy 1
Key Concepts: Scalar products, work, kinetic energy, work-energy theorem.
1.
Questions about the scalar product of two vectors.
a.
Use vectors to prove the law of cosines for a triangle. [Consider the
product C ⋅ C , where C = A + B , and draw the arrows.]
b.
In 1-D kinematics with constant acceleration we had the formula
v 2 = v02 + 2a(x − x0 ) . Take the equations for constant acceleration in 3-D
(page 2 of Kinematics 2) and show that this formula becomes
v 2 = v02 + 2a ⋅ (r − r0 ) .
2.
With modern technology one can measure the acceleration of gravity g directly
by dropping an object and taking a photo of its fall with a rapidly flashing strobe
light. But in Galileo’s time the fall was too rapid to make accurate measurements,
so he used an incline to reduce the acceleration to a more measurable level.
Shown are the modern experiment (a) and Galileo’s (b). In both the object starts
from rest. Assume that in case (b) the object slides down a frictionless incline. (In
fact it was a small ball rolling without slipping.) Neglect air resistance.
h
h
θ
(a)
(b)
Comment on the validity of each statement and explain your reasoning.
a.
The work done by gravity is the same in both cases.
b.
There is no work done by any force other than gravity in either case.
c.
The object will reach the floor with the same speed in both cases.
d.
The power input to the object is the same in both cases.
e.
If the measured acceleration in (b) is a, then g = a/sin θ .
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PHY 53
3.
Summer 2010
A particle of mass m is acted on by a constant net force. It starts from rest at t = 0 .
a.
Show that the power input to the particle is proportional to t.
b.
Show that the kinetic energy of the particle is proportional to t 2 .
c.
Show that the power input is proportional to
x , where x is the distance
traveled since the start.
d.
Show that the distance traveled during any time interval is proportional to
the increase in v 2 during that interval, i.e., Δx  Δv 2 .
4*.
A block of mass m slides down the incline
shown, starting from rest at the top. The
coefficient of kinetic friction is 0.25. Give
answers in terms of m, g and θ . [Assume
tan θ ≥ 0.25 .]
5.
h
θ
a
What is the net force? Ans:
mg(sin θ − 0.25cosθ ) , down the incline.
b.
What is the block’s kinetic energy when it reaches the bottom? Ans:
mgh(1 − 0.25cot θ ) .
c.
What is the total work done by gravity? Ans: mgh .
d.
Calculate the total work done by friction directly, and verify the workenergy theorem. Ans: −0.25mghcot θ .
A block of mass m is being lifted at constant speed by the system
shown. A person pulls on the left end of the rope with tension T.
a.
When the person’s hand moves down distance d, through
what distance does the block move up? Ans: d / 2 .
b.
What is the upward force exerted on the block by the rope,
in terms of T? Ans: 2T .
c.
What is T in terms of m and g? Ans: mg / 2
d.
Compare the power input by the person with the power delivered to the
block. Ans: Same.
2
T
PHY 53
6.
Summer 2010
A ball of mass m attached to a string of length R is rotating
counter-clockwise in a vertical circle as shown. When it is at
the top of the circle at A the tension in the string is zero. Give
answers in terms of m, R and g.
C
a.
What is the ball’s speed at that point? Ans:
b.
How much work is done by gravity between A and B
(the bottom of the circle)? Ans: 2mgR .
c.
How much work is done by the tension in the string as the particle moves
from A to B? Explain. Ans: None.
d.
What is the ball’s speed at B? Ans:
e.
What is the ball’s velocity (magnitude and direction) at C where the string
is horizontal? Ans:
f.
7.
A
Rg .
B
5Rg .
3Rg , vertically upward.
What is the power input at that point? Ans: −mg ⋅ 3Rg .
A block of mass m is attached to a spring of stiffness k, suspended
from a ceiling as shown. Originally it is at rest with the spring
unstretched. It is released and falls. When it has fallen a distance y1 it
is momentarily at rest again.
8.
a.
How much work was done by gravity during the fall, in terms
of m, g and y1 ? Ans: mgy1 .
b.
Calculate the work done by the spring directly from the definition, in
terms of y1 and k. Ans: − 12 ky12 .
c.
Use the work energy theorem to find y1 . Ans: 2mg / k .
d.
What is the total force on the block (magnitude and direction) at this
instant, in terms of m and g? Ans: mg , vertically upward.
When a car speeds up on a level road without the tires slipping, the force that
gives it the acceleration is static friction between the tires (on the drive wheels)
and the road.
a.
Does the increase in kinetic energy of the car come from work done by this
force? Explain.
b.
What is the ultimate source of this energy?
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PHY 53
9.
Summer 2010
A girl sitting on a seat in a bus that is moving with speed V throws a ball of mass
m toward the front of the bus. As seen in the bus the ball is given initial speed v.
a.
What is the increase in kinetic energy of the ball as seen in the bus? Ans:
1
mv 2 .
2
b.
As seen by an observer standing on the curb, what is the increase in
kinetic energy of the ball? [What are the initial and final speeds of the ball,
as the observer sees them?] Ans: 12 m(v 2 + 2vV ) .
c.
Explain the difference. What work is done as seen by the observer, but not
by the girl in the bus?
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