12-*Physics IClass 12Uniform Circular Motion

Rev. 19-Feb-06 GB

12-*Newtons Second Law -Yet Another Review!

Newtons Second Law:

or

The net force and acceleration are always in the same direction because m is a positive number.

Acceleration is the rate of any change in the velocity vector either magnitude (speed) or direction, or both.

Todays lecture and activity will stretch our understanding of acceleration and Newtons Second Law for a special type of motion.

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12-*Uniform Circular Motion

Uniform circular motion means that the object moves in a circle at a constant speed. Some definitions and equations:

T period time to go around exactly once

r radius of circle

v speed (scalar, not vector)

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12-*Acceleration in Circular MotionWhen an object travels in a circle, its velocity is constantly changing (in direction at least).

That means the object has a non-zero acceleration even if it moves at constant speed.

12-*Important Facts AboutVelocity and Acceleration Vectors

Same direction: speeding up.

Opposite directions: slowing down.

Right angles: changing direction, same speed.

12-*What is the Direction of Acceleration?

Since the speed is not changing, only the direction of velocity, acceleration must be always at right angles to velocity. The acceleration vector points inward, toward the center of the circle. This is called centripetal acceleration from Latin forto go to or seek the center.

Like the direction of the velocity vector,the direction of centripetal acceleration is constantly changing as the object moves around the circle.

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The magnitude of centripetal acceleration is given by

.

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12-*Centripetal ForceCentrifugal Force

Centripetal force and centrifugal force are two of the most unnecessary and confusing concepts in introductory physics. Centripetal force is the net force on any object in circular motion. Centrifugal force is a result of measuring the displacement of an object relative to an accelerated (rotating) observer.

In this course, we will never refer to centripetal or centrifugal force. We will refer only to actual physical forces such as gravity.

It will never be correct to answer any question using centripetal force or centrifugal force. Instead, use centripetal acceleration and Newtons Second Law as measured by a non-accelerating observer.

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12-*Example:A Rock on a StringTwirl a 1 kg rock attached to a string in a 1 m radius vertical circle. The speed is 4 m/sec. What forces act on the rock and what are the directions of those forces?

12-*Using Newtons Second Law to Solve Problems - Review

1.Identify all forces acting on the object.

Pushes or PullsFriction (if specified)

GravityNormal (Surface) Forces

2.Choose a coordinate system.

If you know the direction of acceleration, one coordinate axis should be in that direction.

3.Draw a Free-Body Diagram.

We know how to do this now.

4.Express the force vectors in components.

This may require trigonometry.

5.Use Newtons Second Law to write one equation for each direction considered.

We will only consider vertical forces today.

6.Solve the equation(s).

12-*Case A:Rock at the Top of the Circle

The center of the circle is below the rock, so acceleration is down.

1.Forces: Weight (W) down and Tension (T) down.

2.Coordinates: +X down. (Why?)

3.Free-body diagram:

4.X Components: (W) and (T).

5.Second Law: (W) + (T) = m a.

6.Solve:

T = 1 (16/19.8) = 6.2 N.

W = 9.8 N.

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12-*Case B:Rock at the Bottom of the Circle

The center of the circle is above the rock, so acceleration is up.

1.Forces: Weight (W) down and Tension (T) up.

2.Coordinates: +X up. (Why?)

3.Free-body diagram:

4.X Components: (W) and (T).

5.Second Law: (W) + (T) = m a.

6.Solve:

T = 1 (16/1+9.8) = 25.8 N.

W = 9.8 N.

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12-*Normal Force - A ConceptWe Will Use in Todays Activity

Elevator Cab

Normal force is the force generated by a solid object to keep other objects from penetrating into it. As the name implies, the direction of this force is at right angles (normal) to the surface. The physical cause of this force is the stretching of chemical bonds, much like the stretching of a lattice of springs.

Normal force is often equal and opposite to weight, but not always. Consider an elevator cab. How does the normal force compare to weight if the cab is moving at a constant velocity? Accelerating upward? Accelerating downward?

Floor

12-*Class #12Take-Away Concepts

1.Acceleration (or net force) at a right angle to velocity causes a change of direction but not a change of speed.

2.As an object moves around a circle at a constant speed, it accelerates toward the center with magnitude given by

3.Do not try to use centripetal or centrifugal forces to solve a problem. Use centripetal acceleration and Newtons 2nd Law.

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12-*Class #12Problems of the Day

___ 1.Imagine you are riding on a train going around a horizontal curve of radius r at speed v. You are holding an object, of mass m, stationary (relative to the train) in your hand.Let g be the acceleration constant of gravity and a = v2/r.The net force exerted on the object is:

A) mg upward + ma horizontal toward center

B) mg downward + ma horizontal toward center

C) ma horizontal toward center

D) mg upward + ma horizontal away from center

E) mg downward + ma horizontal away from center

F) ma horizontal away from center

G) zero

H) more information is needed to answer this question

12-*Class #12Problems of the Day

2. The space shuttle Atlantis is in a circular orbit 100. km (1.00 x 105 m) above the surface of the earth. (This is an example of uniform circular motion.) At this height, the constant of gravity is g = 9.53 N/kg. The radius of the earth is 6.37 x 106 m. How long (in minutes) does it take for the shuttle Atlantis to complete one orbit?

12-*Activity #12Ferris Wheel Thrill Ride

Objectives of the Activity:

1.More experience with video analysis in LoggerPro.

2.Investigate uniform circular motion

Is the acceleration really directed toward the center?

3.What is the subjective experience of traveling in a rapid vertical circular motion?

12-*Class #12 Optional MaterialCentrifugal Force?

Turn this way.

Feel a force this way.

Centrifugal is from Latin for to flee from the center.

Is centrifugal force a real force?

12-*Accelerated Frames of Reference

Newtons Second Law applies to an inertial reference frame, meaning a reference system for measuring position and time that is not accelerating.

If we wish to use Newtons Second Law in an accelerating reference frame, we need to add extra terms to the equation that can be considered as forces operating on every object that we track using the accelerating reference frame. These are commonly called inertial forces.

Why would we do a crazy thing like using an accelerated reference frame instead of an inertial reference frame? In a way, it is built into human nature to view ourselves as sitting still while the rest of the universe zips by. It is often a convenient way to calculate things as long as we are careful.

12-*The Inertial Forces

#1 #2 #3 #4

Dont worry about understanding the details of this equation. The important thing is that each term represents a different type of inertial force.

1.This term is due to linear (in a straight line) acceleration of the reference frame. For example, in a car if you slam on the brakes (acceleration to the rear), it feels like everything in the car is thrown forward.

2.Centrifugal force this term is due to rotation of the reference frame.

3.Coriolis force this term is due to moving in a rotating reference frame. If you ever tried to walk down the aisle of a train as it rounded a curve, you experienced this force.

4.This term is due to acceleration of the rotation of a reference frame. If you are riding on a merry-go-round, you need to hold on tighter as it starts up.

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12-*DiscussionSwinging a Water Bucket Overhead

Take a sturdy water bucket full of water and swing it in a circle over your head. What will the water do if

1.you swing it quickly?

2.you swing it slowly?

3.you stop it when it is directly overhead?

Can we relate the concepts and math formulas we have talked about today to the transition between situation #1 and situation #2?