1. Rotational Motion and the Law of Gravity
Lecture Notes
Physics 2053
Rotational Motion and the Law of Gravity

2. Rotational Motion and the Law of Gravity
Topics
7-04 Centripetal Acceleration
7-05 Newtonian Gravitation
7-06 Kepler’s Laws
Rotational Motion and the Law of Gravity

3. Centripetal Acceleration
Uniform circular motion: motion in a circle of constant radius at constant speed
Instantaneous velocity is always tangent to circle. v2 v1
Rotational Motion and the Law of Gravity

4. Centripetal Acceleration
Radial Acceleration:
Similar Triangles v2
-v1 v2 Dt Dq r2 Dr Dq
Divide
by Time r1 Dv v1
Divide by time
Uniform
Circular
Motion
Simular
Triangles
Centripetal
Acceleration
Rotational Motion and the Law of Gravity

5. Centripetal Acceleration
In uniform circular motion the acceleration is called the centripetal, or radial, acceleration. It is perpendicular to the velocity and points towards the center of the circle. v ar r
Rotational Motion and the Law of Gravity

6. Rotational Motion and The Law of Gravity
Is it possible for an object moving with a constant speed
to accelerate? Explain.
A) Yes, although the speed is constant, the direction
of the velocity can be changing.
B) No, if the speed is constant then the acceleration
is equal to zero.
C) No, an object can accelerate only if there is a
net force acting on it.
D) Yes, if an object is moving it can experience
acceleration
Rotational Motion and the Law of Gravity

7. Centripetal Acceleration Problem
A jet plane travelling 525 m/s pulls out of a dive by moving in an arc of radius 6.00 km. What is the plane’s acceleration?
Rotational Motion and the Law of Gravity

8. Rotational Motion and The Law of Gravity
An object moves in a circular path at a constant speed.
Compare the direction of the object's velocity and
acceleration vectors.
A) The vectors are perpendicular.
B) Both vectors point in the same direction.
C) The vectors point in opposite directions.
D) The question is meaningless, since the acceleration is zero.
Rotational Motion and the Law of Gravity

9. Rotational Motion and The Law of Gravity
What type of acceleration does an object moving with
constant speed in a circular path experience?
A) free fall
B) constant acceleration
C) linear acceleration
D) centripetal acceleration
Rotational Motion and the Law of Gravity

10. Centripetal Acceleration
For an object to be in uniform circular motion, there must be a net force acting on it.
The radial force
on the ball is
provided by the string v ar Fr
There is no
centrifugal force
acting on the ball
This radial force
is called a
centripetal force
Rotational Motion and the Law of Gravity

11. Centripetal Acceleration
The speed of an object in
Uniform Circular Motion m T r M Mg
Rotational Motion and the Law of Gravity

12. Centripetal Acceleration Problem
A 0.45 kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N, what is the maximum speed the ball can have?
Rotational Motion and the Law of Gravity

13. Centripetal Acceleration
Motion in a vertical circle T mg
The tension in the string
when the ball is at the top. r
Rotational Motion and the Law of Gravity

14. Centripetal Acceleration
Motion in a vertical circle
The tension in the string when
the ball is at the bottom. r T mg
Rotational Motion and the Law of Gravity

15. Centripetal Acceleration Problem
A bucket of mass 2.00 kg is whirled in a vertical circle of
radius 1.10 m. At the lowest point of its motion the tension
in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
Rotational Motion and the Law of Gravity

16. Centripetal Acceleration Problem (con’t)
A bucket of mass 2.00 kg is whirled in a vertical circle of
radius 1.10 m.
(b) How fast must the bucket move at the top of the
circle so that the rope does not go slack?
Rotational Motion and the Law of Gravity

17. Rotational Motion and The Law of Gravity
A pilot executes a vertical dive then follows a semi-circular
arc until it is going straight up. Just as the plane is at its
lowest point, the force on him is
A) less than mg, and pointing up.
B) less than mg, and pointing down.
C) more than mg, and pointing up.
D) more than mg, and pointing down.
Rotational Motion and the Law of Gravity

18. Centripetal Acceleration
Maximum Speed in horizontal turn N r m ax
fmax mg
Rotational Motion and the Law of Gravity

19. Rotational Motion and The Law of Gravity
A car goes around a curve of radius r at a constant speed v.
What is the direction of the net force on the car?
A) toward the curve's center
B) away from the curve's center
C) toward the front of the car
D) toward the back of the car
Rotational Motion and the Law of Gravity

20. Rotational Motion and The Law of Gravity
A car goes around a curve of radius r at a constant speed v.
Then it goes around a curve of radius 2r at speed 2v. What
is the centripetal acceleration of the car as it goes around the
second curve, compared to the first?
A) four times as big
B) twice as big
C) one-half as big
D) one-fourth as big
Rotational Motion and the Law of Gravity

21. Centripetal Acceleration Problem
How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?
Rotational Motion and the Law of Gravity

22. Centripetal Accelerationq
Friction on a banked road v a q f mg q r
Rotational Motion and the Law of Gravity

23. Centripetal Acceleration
Friction on a banked road N v q
When f a
When f q mg
When
No Friction
Rotational Motion and the Law of Gravity

24. Centripetal Accelerationv
Turning a banked curve
with no friction q a q mg
Rotational Motion and the Law of Gravity

25. Rotational Motion and The Law of Gravity
The banking angle in a turn on the Olympic bobsled track
is not constant, but increases upward from the horizontal.
Coming around a turn, the bobsled team will intentionally
"climb the wall," then go lower coming out of the turn.
Why do they do this?
A) to give the team better control, because they are able
to see ahead of the turn
B) to prevent the bobsled from turning over
C) to take the turn at a faster speed
D) to reduce the g-force on them
Rotational Motion and the Law of Gravity

26. Centripetal Acceleration
Weight
Reading on scale
is the normal force N mg
Scale
Rotational Motion and the Law of Gravity

27. Centripetal Acceleration
Apparent Weight at the Earth’s Surface
At the North Pole: NN mg
At the Equator: v NE mg
Rotational Motion and the Law of Gravity

28. Centripetal Acceleration
A space station is in the shape of a hollow ring 450 m
in diameter. Gravity is simulated by rotating the ring.
Find the speed in revolutions per minute needed in
order to simulate the Earth’s gravity. R v N
Rotational Motion and the Law of Gravity

29. Centripetal Acceleration
The speed in revolutions per minute v N
R = 225 m
Rotational Motion and the Law of Gravity

30. Newtonian Gravitation
Gravitational Force:
Gravitational Force is the mutual force of attraction between any two objects in the Universe. m F R M
Universal Gravitational Constant
Rotational Motion and the Law of Gravity

31. Newtonian Gravitation
Gravitational Potential Energy associated with an object of mass m at a distance r from the center of the Earth is r m ME
Rotational Motion and the Law of Gravity

32. Newtonian Gravitation
Escape velocity
An object projected upward from the Earth’s surface with a large enough speed will soar off into space and never return.
This speed is called the Earth’s escape velocity.
Rotational Motion and the Law of Gravity

33. Newtonian Gravitation Problem
Calculate the acceleration due to gravity on the Moon. The Moon’s radius is 1.74 x 106 m and its mass is 7.35 x 1022 kg.
Rotational Motion and the Law of Gravity

34. Newtonian Gravitation Problem
A hypothetical planet has a mass 1.66 times that of Earth, but the same radius. What is g near its surface?
Rotational Motion and the Law of Gravity

35. Newtonian Gravitation
Speed of a Satellite v m F R M
Rotational Motion and the Law of Gravity

36. Rotational Motion and The Law of Gravity
Two planets have the same surface gravity, but planet B has twice the radius of planet A. If planet A has mass m, what is the mass of planet B?
Rotational Motion and the Law of Gravity

37. Newtonian Gravitation Problem
A certain neutron star has five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster.
Rotational Motion and the Law of Gravity

38. Newtonian Gravitation
The satellite is kept in orbit by its speed – it is continually falling, but the Earth curves from underneath it.
Without
gravity
With
gravity
Earth
Rotational Motion and the Law of Gravity

39. Rotational Motion and The Law of Gravity
Compared to its mass on the Earth, the mass of an object
on the Moon is
A) the same.
B) less.
C) more.
D) half as much.
Rotational Motion and the Law of Gravity

40. Newtonian Gravitation Problem
Calculate the force of Earth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kg.
Rotational Motion and the Law of Gravity

41. 1. All planets move in elliptical orbits with the
Kepler’s Laws
Kepler’s Three Laws:
1. All planets move in elliptical orbits with the
Sun at one of the focal points.
Rotational Motion and the Law of Gravity

42. 2. A line drawn from the Sun to any planet sweeps
Kepler’s Laws
Kepler’s Three Laws:
2. A line drawn from the Sun to any planet sweeps
out equal areas in equal time intervals.
Area 1
Area 2
Area 1 = Area 2
Rotational Motion and the Law of Gravity

43. 3. The square of the orbital period of any planet is
Kepler’s Laws
Kepler’s Three Laws:
3. The square of the orbital period of any planet is
proportional to the cube of the average distance
from the planet to the Sun. T R
Rotational Motion and the Law of Gravity

44. Kepler’s Laws T r
Rotational Motion and the Law of Gravity

45. END