Understanding Circular Motion and Newton's Laws: A Comprehensive Guide

1. 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 1 rad = = 57.3° • 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. 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 Relationship Between Angular and Linear Quantities

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