Download CM-Conservation of Energy

AP PHYSICS
NAME___________________________
CH. 7 PE AND ENERGY CONSERVATION
DATE______________PERIOD________
CIRCULAR MOTION & ENERGY
CONSERVATION
1.
A car in an amusement park ride rolls without friction around the track as shown below. It
A the car as a
starts from rest at point A at a height h above the bottom of the loop. Treat
particle.
B
C
h
R
a. What is the minimum value of h (in terms of R) such that the car moves around the loop without
falling off at the top (point B)?
b. If h = 3.50R and R = 20.0 m, compute the speed, radial acceleration, and tangential acceleration
of the passengers when the car is at point C, which is at the end of a horizontal diameter. Show
these acceleration components in the diagram above approximately to scale.
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2. A skier starts at the top of a very large, frictionless snowball, with a very small, initial speed, and
skis straight down the side. At what point does she lose contact with the snowball and fly off at
a tangent? That is, at the instant she loses contact with the snowball, what angle α does a radial
line from the center of the snowball to the skier make with the vertical?
α
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3. An object of mass m is released from rest at a height h above the surface of a table. The
object slides along the inside of the loop-the-loop track consisting of a ramp and a circular
loop of radius R shown in the figure. Assume that the track is frictionless. When the object is
at the top of the track it pushes against the track with a force equal to three times it’s weight.
What height was the object dropped from?
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4. A ring of mass m1 hangs from a thread, and two identical beads of mass m2 slide on it
without friction. The beads are released simultaneously from the top of the ring from rest
(actually they need a very small initial speed but this can be ignored) and slide down
opposite sides. Assume m2 > (3/2) m1. The ring will start to rise when the beads reach a
critical angle θc with respect to the vertical.
a. Draw free body force diagrams for the ring and the beads. What direction is the force of the
bead on the ring pointing? Does it change as the bead moves? Can you still proceed with an
analysis using Newton’s Second Law if you are not sure which way this force points? Try to
find a physical explanation for the direction of this force. What is the condition that the ring
just starts to rise?
b.
Does the mechanical energy change between when the beads are released and the ring just
starts to rise? Write down an equation that describes the changes in the mechanical energy.
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5. A small object of mass m =0.2 kg is placed at the top of a large sphere of radius R =0.5 m
resting on the ground. The object is given a negligibly small velocity so that it starts to slide
down the sphere. Assume the surface of the sphere is frictionless and the sphere is fixed to
the surface of a table. In this problem, you will try to find where the object hits the ground.
a) Briefly describe how you intend to model the motion. Here are some questions which should
help guide your thinking. Is there any special condition(s) that describe when the object
leaves the sphere? Does the normal force do any work on the object? Does the principle of
Conservation of Energy replace Newton’s Second Law in the radial direction or are they
independent?
b) At what angle will the mass leave the sphere?
c) What is the velocity of the mass when it just leaves the sphere?
d) How far from the bottom of the sphere will the mass hit the ground?
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6. A ball of negligible radius and mass m = 0.1 kg hangs from a string of length l = 0.5
m. It is hit in such a way that it then travels in a vertical circle (with negligible loss of
energy). The initial speed of the ball after being struck is v = 7.0 m/s.
a) Find the speed of the ball at the top of the circle.
b) b) Find the tension in the string when the ball is at the top of the circle.
0
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7. A bead of mass M slides on a smooth wire that is bent in a circle of radius R. It is released at
the top of the circular part of the wire (point a in the figure) with a negligibly small velocity.
a.
Find the normal force of the wire on the bead at point b (even with the center of the
circle).
b. Find the normal force at point c (at the bottom of the circle).
c. Find the height H of point d where the mass will reverse direction.
d
a
b
H
c
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