VIDEO LECTURE FORCES AND CIRCULAR MOTION

Mazur Section 11.2 Objectives

2

¨ Determine the force necessary to keep an object moving in a circle

¨ Give examples of various causes of centripetal force, and analyze the constituent forces

¨ Describe “fictitious forces” and non-inertial frames of reference Centripetal force

3

¨ To keep an object in a circular motion, the net force must be pointed toward the center of the circle.

¨ Any form of such force (normal force, tension, gravity, etc) is called “centripetal force.”

¨ Centripetal force is just a name, so you should NEVER label a “centripetal force” in a FBD. Centripetal force observations

4

¨ At the same (translational) speed and mass, a larger force makes for a smaller radius of revolution

¨ Moving at the same (translational) speed, the object with a smaller radius has larger (translational) acceleration Acceleration & force

5

¨ Centripetal acceleration:

¨ Tangential acceleration:

¤ at is zero if a uniform circular motion.

¤ Non-zero at changes speed. ¨ Apply Newton’s 2nd Law: Direction of travel

6

¨ Imagine a rock is whirled around in a uniform circular motion by a string.

¨ The tension in the string supplies the centripetal force necessary for the rock to follow the circular path.

¨ The velocity of the rock is always tangent to the path.

¨ If the string breaks, the rock will move along the straight line, tangent to the circular path. Recall Newton’s first law.

no Centripetal force practice

7

¨ A cosmonaut is in a spacecraft orbiting Earth at an altitude h = 600 km with a speed v = 7.6 km/s. His mass is m = 80 kg. Earth’s radius is 6400 km a) What is his acceleration? b) What (centripetal) gravitational force does Earth exert on the cosmonaut? Sources of centripetal force example

8

¨ A bob suspended from a string moves in a horizontal circle at constant speed. ¨ No vertical acceleration ¨ Centripetal acceleration is due to the horizontal component of the tension. Conical pendulum continued

9

¨ You cannot have perfectly horizontal conical pendulum. If q = 90°, v must be infinite. Conical pendulum practice

10

¨ The length of a string of a conical pendulum is L = 50.0 cm, and the mass of the bob is m = 0.25 kg. a) Find the angle between the string and the horizontal when the tension in the string is six times the weight of the bob. b) Under those conditions, what is the period of the pendulum? Unbanked curves

11

¨ On a unbanked road, static friction by the road on the car provides the centripetal force necessary for the car to follow a curve. ¨ It is static friction because no slipping occurs at the point of contact between road and tires. Unbanked curves: 2

12

¨ The maximum speed the car can go is limited by the maximum static friction the road can exert on the tire. Unbanked curve example

13

¨ A bicyclist travels at a constant speed of 9.00 m/s in a circle of radius 25.0 m on a flat ground. The combined mass of the bicycle and rider is 85.0 kg. ¤ Calculate the magnitude of the force of friction exerted by the road on the bicycle. ¤ What is the coefficient of the road and rubber? Banked curves

14

¨ On a banked road, the normal force of the road will have a component in the centripetal direction. ¨ Banking is useful, e.g., for an icy road. ¨ The banking angle is usually chosen so that no friction is needed for a car to complete the curve at the specified speed.

n Banked curves: 2

15

¨ Banking angle: n

Banking angle Loop-the-Loop

16

¨ If you are going fast enough, the normal force on the bike is non- zero, i.e, the bike is touching the track. ¨ If you are barely making it, the normal force at the top is zero.

At the top

mg

n Loop-the-loop example

17

¨ The radius of curvature of the track at the top of a loop-the-loop on a roller-coaster ride is r = 12.0 m. At the top of the loop, the force that the seat exerts on a passenger of mass m is 0.40mg. How fast is the roller-coaster car moving as it moves through the highest point of the loop? ROTATING REFERENCE FRAMES AND FICTITIOUS FORCES

Mazur Section 11.1, 11.4 (comic by xkcd) Rotating frames of reference

19

¨ Newton’s laws of motion describe observations made in an inertial frame of reference. ¤ An inertial frame of reference moves at constant velocity.

¨ To an observer in a non-inertial frame of reference, Newton’s laws appear to be violated. ¤ A non-inertial frame of reference is accelerating. Rotating frames of reference

20 Acceleration in rotating coordinates

21

¨ Acceleration of the bob has components in the radial and tangential directions.

¨ The radial component:

¨ The tangential component: Fictitious forces

22

¨ For an observer in a non-inertial frame of reference, an object can accelerate without any external net force applied to it.

¨ We call such an apparent force that causes the object to accelerate a fictitious force.

¨ Real forces are always interactions between two objects, but there is no second object for a fictitious force.

¨ Fictitious forces appear to act in the opposite direction from the direction of the acceleration of the non-inertial frame.

¨ This is a good reason to always draw free body diagrams in non-accelerating frames of reference Speeding up

23

¨ When a car is speeding up, you (in a non-inertial frame) feel as if you are pushed back against the seat.

¨ This apparent “force” is a fictitious force.

¨ There is no force actually pushing you back.

¨ When the car is speeding up, the friction on the road pushes the tires forward, accelerating the car.

¨ If the car seat does not supply enough friction on you, you cannot accelerate as much as the car, and you would lag behind. “Centrifugal force” from turning

24

¨ When a car is turning a corner on a flat road, you feel as if you are pushed in the direction away from the center of the curvature.

¨ There is no force actually pushing you outward.

¨ But this fictitious outward force is called “centrifugal force.” “Centrifugal force” from turning: 2

25

¨ The friction of the road supply the centripetal force on the car, but not on you.

¨ So, you tend to keep going in a straight path. Remember “inertia.” 1. Observation made by you in the car, non-inertial frame of reference. 2. Observation made by an observer on the ground, inertial frame of reference. Coriolis force from rotation

26

¨ A Coriolis force is an apparent force on an object viewed in a rotating frame of reference.

¨ The “Coriolis force” due to the rotation of Earth is responsible for rotations of hurricanes and large- scale ocean currents. Next video

27

¨ Next video will introduce the moment of inertia (rotational inertia in the book)