Download Chapter 05 Solutions

5 Circular Motion, the Planets, and Gravity
Answers to Questions
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
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Q11
Q12
Q13
No, centripetal acceleration is a change in direction, as in orbital motion.
a. The velocity of a body moving in a curve is continuously changing because of the change in direction
even though the speed is constant.
b. Yes. Any change in velocity is acceleration.
The car with the higher speed experiences a larger change in velocity. The change in velocity for a body
moving in a curved path at constant speed depends on the speed and the angle between the direction of
the final and initial velocity vectors. The car with the higher speed is taking the curve in a shorter interval of
time.
The change in velocity is greater for the curve with the smaller radius. Since the distance and speed were
the same for both cases, it comes down to comparing rate of change in angular direction.
Because both the magnitude and direction of the ball’s velocity increase, the acceleration increases
proportional to the product of the two changes.
Path 3, in the direction of the tangent to point A. Neglecting gravity, the body would move in the direction it
was moving when the force disappeared, in accordance with the first law.
There is a force radially inward due to the tension in the string that supplies the centripetal force needed for
this motion. There is also the gravitational force due to Earth’s proximity.
Only if the twirling is done in a region of space not subject to gravitational force of any body. But of course
in that case ‘horizontal’ is arbitrary. As long as there is a pull of gravity on the string, there will be a vertical
component to the force.
a.
b. The direction of the net force is radially inward to the center of the curve.
Yes. Above some maximum speed there will not be sufficient friction (static) to provide the necessary
centripetal force for circular motion, and the car will slide. The maximum speed will depend on the nature of
the surface and the radius of curvature of the road.
Yes. Circular motion requires centripetal force. For the banked roadway, the normal reaction force of the
road on the car can provide a horizontal component yielding the correct centripetal force for a given speed.
At this speed, no friction will be required.
For a body in a vertical circle there will be two contributions to the force acting on a body: its weight plus
the tension in the string. The vector sum of these forces must equal the mass times the constant value of
the centripetal acceleration. At the lowest point the tension is the greatest and equals the sum of the
weight plus the mass times acceleration.
The rider’s weight (downward) is the largest force.
N
T (motor)
f
mg
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The Copernican model assumed that the planets moved in simple orbits about the sun. The Ptolemaic
model had to assume a more complicated motion involving a planet moving in a small circle while the
center of that circle moved in a larger circle about the Earth.
In Ptolemy's view, everything revolved about the Earth. The Earth was stationary and the sun and stars
revolved about it, as well as the planets in their epicycles.
All objects around us are moving at the same speed, including the air. Hence, there is no sense of motion
relative to our surroundings.
Copernicus, like the ancient astronomers, believed that nature is perfect and that orbits should be circles,
since these are the most perfect curves. Although Kepler was a mystic also seeking perfection, he realized
that the observational data required the orbits to be ellipses rather than circles.
The ellipse becomes less elongated as the foci are brought closer together and becomes a circle when the
two focal points coincide.
A planet in an elliptical orbit about the sun moves fastest when it is nearest the sun. This is because an
equal area of the ellipse is traversed in the same time interval at both aphelion and perihelion (as well as
for points in between).
From Newton’s third law we know that the interaction of two masses results in equal but oppositely directed
forces.
There is a net gravitational force acting on the Earth. The sun exerts the largest force, the moon a lesser
force, and the more distant heavenly bodies much smaller forces.
There will be a net force acting on m 2 toward m1. The third mass exerts a force of attraction to the right, but
since it is farther away, that force is less than the force exerted by m 1 to the left.
Since the gravitational force of attraction varies as 1/r 2, doubling the distance between them will result in
1/4 of the original force acting between them.
No. There are no stars between the Earth and the moon. (Maybe blinking lights of a passing jet?)
The new moon rises in the morning and sets at dusk.
The first quarter moon rises and sets about six hours after the sun, while the third quarter moon rises and
sets about six hours before the Sun. (Noon and midnight are the rising and setting times depending on
lunar phase.)
New moon occurs when the moon passes between the Earth and the sun. Thus the side of the moon
facing us is not illuminated (except possibly by earthshine—the light originating from Earth being reflected
by the moon).
Solar eclipses occur during the new moon and only when both sun and moon are aligned with the same
lunar node.
Because centripetal force is holding it in orbit, it is not ‘standing still’. This centripetal force is a result of its
acceleration around Earth. It happens that its orbit takes the same amount of time as one diurnal rotation of
Earth.
Kepler's third law applies to all satellite motion but in the case of satellites orbiting Earth the mass of the
sun is replaced by the mass of the Earth so that there is a different ratio of T 2/r3.
Because the moon moves relative to the Earth it takes about 25 hours for the moon to return to the same
point in the sky, so there are two high tides every 25 hours.
The center of the Earth is closer to the Moon than the water at the far side of the Earth. Thus the center of
the Earth is subject to a greater influence from the Moon than is the water.
No. The phenomenon of tides comes about because of the difference between the force acting on the
water at the surface and the force acting on the center of the rigid Earth.
Answers to Exercises
E1
E2
E3
E4
E5
E6
31.3 m/s2
16 m/s2
1m
4 times
1N
a. 10.0 m/s2
b. 12 kN
2
E7
E8
E9
E10
E11
E12
E13
E14
E15
E16
a. 18.22 m/s2
b. 18.2 kN
a. 5.33 m/s2
b. 373.3 N
365 : 1
180 N
0.04 N
2.67 x 10-6 N
.56 N (4 times greater)
30 lb
299.7 lb
a. 2:35 AM
b. 8:47 AM, 9:12 PM
Answers to Synthesis Problems
SP1
a.
b.
c.
d.
26.67 m/s2
5.34 N
1.96 N
Instructor will show students.
SP2
a.
b.
c.
d.
e.
9.42 m/s
7.4 m/s2
2.96 x 102 N is the necessary centripetal force; Yes.
96 N
If the rider lets go of the safety bar the rider will fly out at a trajectory tangent to the ferris wheel’s
rotation (parallel to the ground). At that moment gravity will also accelerate the rider to the ground.
SP3
a.
b.
c.
d.
e.
10.4 m/s2
9.36 kN
8.82 kN
Diagram below; about 9.1 kN
2363 N; No. It is not sufficient
3
SP4
a. 18.8 m/s
b. 118 m/s2 Very large, 12 times.
c. 7106 N
SP5
a.
b.
c.
d.
SP6
a. 7.5%
b. The moon advances in its orbit by 13.2o/day while the Earth advances 1.0o/day for a net of 12.2o/day.
In 27.3 days the moon will have an apparent advance of 333o, so it is not yet a full moon.
c. The moon has 27o to go to full moon. This takes about 2.2 days more.
3.53 x 1022 N
2.01 x 1020 N
175/1; no
4.34 x 1020 N, Not much, but some. The sun’s force keeps the moon in its annual orbit about the sun as
it moves along with the Earth.
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