1. Circular Motion and Other Applications of Newton’s Laws
Unit 5
Circular Motion
and
Other Applications of Newton’s Laws

2. Uniform Circular Motion
A force, Fr , is directed toward the center of the circle
This force is associated with an acceleration, ac
Applying Newton’s Second Law along the radial direction gives

3. Uniform Circular Motion, cont
A force causing a centripetal acceleration acts toward the center of the circle
It causes a change in the direction of the velocity vector
If the force vanishes, the object would move in a straight-line path tangent to the circle

4. Centripetal Force
The force causing the centripetal acceleration is sometimes called the centripetal force
This is not a new force, it is a new role for a force
It is a force acting in the role of a force that causes a circular motion

5. Horizontal (Flat) Curve
The force of static friction supplies the centripetal force
The maximum speed at which the car can negotiate the curve is
Note, this does not depend on the mass of the car

6. Banked Curve These are designed with friction equaling zero
There is a component of the normal force that supplies the centripetal force

7. Non-Uniform Circular Motion
The acceleration and force have tangential components
Fr produces the centripetal acceleration
Ft produces the tangential acceleration
SF = SFr + SFt

8. Vertical Circle with Non-Uniform Speed
The gravitational force exerts a tangential force on the object
Look at the components of Fg
The tension at any point can be found

9. Loop-the-Loop This is an example of a vertical circle
At the bottom of the loop (b), the upward force experienced by the object is greater than its weight

10. Loop-the-Loop, Part 2
At the top of the circle (c), the force exerted on the object is less than its weight

11. Top and Bottom of Circle
The tension at the bottom is a maximum
The tension at the top is a minimum
If Ttop = 0, then

12. Rigid Object A rigid object is one that is nondeformable
The relative locations of all particles making up the object remain constant
All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible

13. Angular Position Axis of rotation is the center of the disc
Choose a fixed reference line
Point P is at a fixed distance r from the origin

14. Angular Position, 2
As the particle moves, the only coordinate that changes is q
As the particle moves through q, it moves though an arc length s.
The arc length and r are related:
s = q r

15. Radian This can also be expressed as
q is a pure number, but commonly is given the artificial unit, radian
One radian is the angle subtended by an arc length equal to the radius of the arc

16. Conversions Comparing degrees and radians
Converting from degrees to radians
 [rad] = [degrees]

17. Angular Displacement
The angular displacement is defined as the angle the object rotates through during some time interval
This is the angle that the reference line of length r sweeps out

18. Instantaneous Angular Speed
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

19. Angular Speed, final Units of angular speed are radians/sec
rad/s or s-1 since radians have no dimensions
Angular speed will be positive if  is increasing (counterclockwise)
Angular speed will be negative if  is decreasing (clockwise)

20. Instantaneous Angular Acceleration
The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0

21. Angular Acceleration, final
Units of angular acceleration are rad/s2 or s-2 since radians have no dimensions
Angular acceleration will be positive if an object rotating counterclockwise is speeding up
Angular acceleration will also be positive if an object rotating clockwise is slowing down

22. Angular Motion, General Notes
When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration
So q, w, a all characterize the motion of the entire rigid object as well as the individual particles in the object

23. Directions, details
Strictly speaking, the speed and acceleration (w, a) are the magnitudes of the velocity and acceleration vectors
The directions are actually given by the right-hand rule

24. Rotational Kinematics
Under constant angular acceleration, we can describe the motion of the rigid object using a set of kinematic equations
These are similar to the kinematic equations for linear motion
The rotational equations have the same mathematical form as the linear equations

25. Rotational Kinematic Equations

26. Comparison Between Rotational and Linear Equations

27. Relationship Between Angular and Linear Quantities
Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion

28. Speed Comparison
The linear velocity is always tangent to the circular path
called the tangential velocity
The magnitude is defined by the tangential speed

29. Acceleration Comparison
The tangential acceleration is the derivative of the tangential velocity

30. Speed and Acceleration Note
All points on the rigid object will have the same angular speed, but not the same tangential speed
All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration
The tangential quantities depend on r, and r is not the same for all points on the object

31. Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration
Therefore, each point on a rotating rigid object will experience a centripetal acceleration

32. Resultant Acceleration
The tangential component of the acceleration is due to changing speed
The centripetal component of the acceleration is due to changing direction
Total acceleration can be found from these components

33. Rotational Motion Example
For a compact disc player to read a CD, the angular speed must vary to keep the tangential speed constant (vt = wr)
At the inner sections, the angular speed is faster than at the outer sections

34. Moment of Inertia The resistance of matter to rotational motion.
The dimensions of moment of inertia are ML2 and its SI units are kg.m2
We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass Dmi

35. Moments of Inertia of Various Rigid Objects

36. Torque
Torque, t, is the tendency of a force to rotate an object about some axis
Torque is a vector
t = r F sin f = F d
F is the force
f is the angle the force makes with the horizontal
d is the moment arm (or lever arm)

37. Torque, cont
The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force
d = r sin 

38. Torque, final
The horizontal component of F (F cos f) has no tendency to produce a rotation
Torque will have direction
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

39. Net Torque
The force F1 will tend to cause a counterclockwise rotation about O
The force F2 will tend to cause a clockwise rotation about O
St = t1 + t2 = F1d1 – F2d2

40. Torque vs. Force Forces can cause a change in linear motion
Described by Newton’s Second Law
Forces can cause a change in rotational motion
The effectiveness of this change depends on the force and the moment arm
The change in rotational motion depends on the torque

41. Torque Units The SI units of torque are N.m
Although torque is a force multiplied by a distance, it is very different from work and energy
The units for torque are reported in N.m and not changed to Joules

42. Torque and Angular Acceleration
Consider a particle of mass m rotating in a circle of radius r under the influence of tangential force Ft
The tangential force provides a tangential acceleration:
Ft = mat

43. Torque and Angular Acceleration, Particle cont.
The magnitude of the torque produced by Ft around the center of the circle is
t = Ft r = (mat) r
The tangential acceleration is related to the angular acceleration
t = (mat) r = (mra) r = (mr 2) a
Since mr 2 is the moment of inertia of the particle,
t = Ia
The torque is directly proportional to the angular acceleration and the constant of proportionality is the moment of inertia

44. Motion in Accelerated Frames
A fictitious force results from an accelerated frame of reference
A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force

45. “Centrifugal” Force
From the frame of the passenger (b), a force appears to push her toward the door
From the frame of the Earth, the car applies a leftward force on the passenger
The outward force is often called a centrifugal force
It is a fictitious force due to the acceleration associated with the car’s change in direction

46. “Coriolis Force”
This is an apparent force caused by changing the radial position of an object in a rotating coordinate system
The result of the rotation is the curved path of the ball

47. Fictitious Forces, examples
Although fictitious forces are not real forces, they can have real effects
Examples:
Objects in the car do slide
You feel pushed to the outside of a rotating platform
The Coriolis force is responsible for the rotation of weather systems and ocean currents

48. Fictitious Forces in Linear Systems
The inertial observer (a) sees
The noninertial observer (b) sees

49. Fictitious Forces in a Rotating System
According to the inertial observer (a), the tension is the centripetal force
The noninertial observer (b) sees