Circular Motion Kinematics of Uniform Circular Motion Dynamics of Uniform Circular Motion Car Rounding a Curve

Motion in Chapter 4 was always in a straight line. In this chapter, we introduce circular motion. Newton’s laws still apply. New chapter, old problems. The biggest difference involves the acceleration… We’ll get to gravitation after exam 2. Chapter 5 Circular Motion Gravitation

Do you remember the equations of kinematics? There are analogous equations for rotational quantities. You will see them later in the course. This section is somewhat mis-titled. I believe our starting point for circular motion best involves forces (dynamics). Consider an object moving in a straight line. ViVi 5.1 Kinematics of Uniform Circular Motion

A force F applied parallel to the direction of motion for a time  t increases the magnitude of velocity by an amount a  t, but does not change the direction of motion. ViVi F  V=a  t A force applied parallel to an object’s velocity vector increases the object’s speed. A force F applied perpendicular to the direction of motion for a time  t changes the direction of the velocity vector. ViVi F  V=a  t In the limit  t  0, the length of the velocity vector does not change.

ViVi F  V=a  t VfVf A force applied perpendicular to an object’s velocity vector instantaneously changes the direction of the velocity vector, but not the object’s speed. If the applied force is always perpendicular to the velocity vector, the object constantly changes direction, but never speeds up or slows down. The object follows a circular path!

Summary and consequences: A force applied perpendicular to an object’s velocity vector instantaneously changes the direction of the velocity vector, but not the object’s speed. If you apply a force parallel to the velocity vector you can only change an object’s speed, not its direction of motion. If you apply a force perpendicular to an object’s velocity vector, you will change its direction of motion BUT NOT ITS SPEED! A force applied parallel to an object’s velocity vector increases the object’s speed.

An example of the latter is circular motion.  A ball tied to the end of a string and “whirled” around.  A child on a merry-go-round.  A car rounding a circular curve.  The earth orbiting the sun (approximately).  The moon orbiting the earth (approximately).

Circular motion “toys:” Position, velocity, and acceleration in uniform circular motion. Circular motion review. Car rounding a bend. Car on a circular banked trackCar on a circular banked track (click the “play” button, then click “initialize/reset.” Hot Wheels Track:

An object moving in a circle with constant speed is said to undergo uniform circular motion. A ball on a string: The instantaneous velocity is tangent to the path of motion (OK to “attach” velocity to object—this is not a free-body diagram). v The instantaneous acceleration is perpendicular to the velocity vector. The ball is accelerated because its velocity constantly changes. If the motion is uniform circular, the acceleration is towards the center of the circle; i.e., the acceleration is “radial” or “centripetal.” a

The force that accelerates the ball is the tension in the string to which it is attached. T The centripetal force due to the string gives rise to a centripetal (also called radial) acceleration. If the ball moves uniformly in a circle, both the force and acceleration continually change direction, so that they always point to the center of the circle. acac Your book shows that

Note: if the motion is not uniform (the speed changes or the radius of the circle changes) there will also be a tangential acceleration. We will not worry about that case in Physics 31. For the problems from this chapter, you need to recall the definitions of frequency, period, and know how to use the fact that that an object moving in a circle with constant speed has speed given by v = 2  r / T. Note that this is not on your OSE sheet. Perhaps it should be. circumference = 2  r speed = distance time

Circular motion: ball on a string, horizontal circle.

Newton’s Laws still apply. The OSE sheet contains a variation on Newton’s second law where the subscript “r” stands for “radial.” You may also use the subscript “c” (“centripetal”). 5.2 Dynamics of Uniform Circular Motion

T acac Choose an axis parallel to the instantaneous acceleration vector. The direction of the y-axis is irrelevant here. x Example. Suppose the ball (mass m) in the example I gave in the previous sections moves with a constant speed V. What is the tension in the string?

“Isn’t there an outward force, pulling the ball out?” NO! T acac x “How about a velocity force?” NO, NO, NO! V

“What about the centrifugal force I feel when my car goes around a curve at high speed?” There is no such thing* as centrifugal force! “But I feel a force!” You feel the centripetal force of the door pushing you towards the center of the circle of the turn! Your confused brain interprets the effect of Newton’s first law as a force pushing you outward. *Engineers talk about centrifugal force when they attach a coordinate system to an accelerated object. In order to make Newton’s Laws work, they have to invent a pseudoforce, which they call “centrifugal force.”

I’ll work this example if there is time. Example 5-2. The moon’s nearly circular orbit about the Earth has a radius of 384,000 km and a period T of 27.3 days. Determine the acceleration of the moon toward the earth. There is a web page for this text, with lots of good material. Here is the Chapter 5 starting page. Here

When you are riding in a car rounding a sharp curve, you may think you feel a centrifugal force. We can use our expert techniques and equation for centripetal force to solve car-rounding-curve problems. 5.3 A Car Rounding a Curve In reality, your body “wants” to continue in a straight line, and you feel the seat force accelerating you along the curved path the car follows.

But first—this may seem strange, but the part of a rolling tire that is in contact with the road is instantaneously at rest. The friction force between rolling tire and road is thus a static friction force. We’ll explore this in more detail later.

Example. Calculate the minimum coefficient of friction between tires and road that will allow a car traveling with a speed V to round a flat curve of radius R without sliding.

Example 5-8. For a car traveling with speed V around a curve of radius R, determine the angle at which the road should be banked so no friction is required.

Example. What is the maximum safe speed around a banked curve of radius R and incline  with road coefficient of friction  ? See Physics 23 Lecture 8.Physics 23 Lecture 8