The Law of Gravity and Rotational Motion

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1 The Law of Gravity and Rotational Motion
Chapters 7 & 8 The Law of Gravity and Rotational Motion

2 Newton’s Law of Universal Gravitation
Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

3 Universal Gravitation, 2
G is the constant of universal gravitational G = x N m² /kg² This is an example of an inverse square law

4 Universal Gravitation, 3
The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1 The forces form a Newton’s third law action-reaction

5 Universal Gravitation, 4
The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center This is called Gauss’ Law

6 Gravitation Constant Determined experimentally Henry Cavendish
1798 The light beam and mirror serve to amplify the motion

7 Applications of Universal Gravitation
Acceleration due to gravity g will vary with altitude

8 Kepler’s Laws All planets move in elliptical orbits with the Sun at one of the focal points. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

9 Kepler’s Laws, cont. Based on observations made by Brahe
Newton later demonstrated that these laws were consequences of the gravitational force between any two objects together with Newton’s laws of motion

10 Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus. Any object bound to another by an inverse square law will move in an elliptical path Second focus is empty

11 Kepler’s Second Law A line drawn from the Sun to any planet will sweep out equal areas in equal times Area from A to B and C to D are the same

12 Kepler’s Third Law The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. For orbit around the Sun, K = KS = 2.97x10-19 s2/m3 K is independent of the mass of the planet

13 Kepler’s Third Law

14

15 The Radian The radian is a unit of angular measure
The radian can be defined as the arc length s along a circle divided by the radius r

16 More About Radians Comparing degrees and radians
Converting from degrees to radians

17 Angular Displacement Axis of rotation is the center of the disk
Need a fixed reference line During time t, the reference line moves through angle θ

18 Rigid Body Every point on the object undergoes circular motion about the point O All parts of the object of the body rotate through the same angle during the same time The object is considered to be a rigid body This means that each part of the body is fixed in position relative to all other parts of the body

19 Angular Displacement, cont.
The angular displacement is defined as the angle the object rotates through during some time interval The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement

20 Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

21 Angular Speed, cont. The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero Units of angular speed are radians/sec rad/s Speed will be positive if θ is increasing (counterclockwise) Speed will be negative if θ is decreasing (clockwise)

22 Average Angular Acceleration
The average angular acceleration of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

23 Angular Acceleration, cont
Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

24 Angular Acceleration, final
The sign of the acceleration does not have to be the same as the sign of the angular speed The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

25 Analogies Between Linear and Rotational Motion

26 Force vs. Torque Forces cause accelerations
Torques cause angular accelerations Force and torque are related

27 Torque The door is free to rotate about an axis through O
There are three factors that determine the effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied

28 Torque, cont Torque, t, is the tendency of a force to rotate an object about some axis t = r F t is the torque F is the force symbol is the Greek tau r is the length of the position vector SI unit is N.m

29 Direction of Torque Torque is a vector quantity
The direction is perpendicular to the plane determined by the position vector and the force If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

30 Multiple Torques When two or more torques are acting on an object, the torques are added As vectors If the net torque is zero, the object’s rate of rotation doesn’t change

31 General Definition of Torque
The applied force is not always perpendicular to the position vector The component of the force perpendicular to the object will cause it to rotate

32 General Definition of Torque, cont
When the force is parallel to the position vector, no rotation occurs When the force is at some angle, the perpendicular component causes the rotation

33 General Definition of Torque, final
Taking the angle into account leads to a more general definition of torque: t = r F sin q F is the force r is the position vector q is the angle between the force and the position vector

34 Lever Arm The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin q

35 Right Hand Rule Point the fingers in the direction of the position vector Curl the fingers toward the force vector The thumb points in the direction of the torque