Circular Motion PPP

Characteristics of Characteristics of Circular MotionCircular Motion

Q&A using animations and questions from Q&A using animations and questions from The Physics Classroom atThe Physics Classroom athttp://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/u6l1c.htmlhttp://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/u6l1c.html

Acceleration Acceleration Characteristics for Characteristics for

Circular MotionCircular Motion

An object moving in uniform An object moving in uniform circular motion is moving in a circular motion is moving in a

circle with a uniform or constant circle with a uniform or constant speed. speed.

Explain why there is Explain why there is acceleration in circular motion acceleration in circular motion

if the SPEED is Constant.if the SPEED is Constant. An accelerating object is an object which is An accelerating object is an object which is

changing its velocity. changing its velocity. Since velocity is a vector which has both Since velocity is a vector which has both

magnitude and direction, a change in either magnitude and direction, a change in either the magnitude or the direction constitutes a the magnitude or the direction constitutes a change in the velocity. change in the velocity.

For this reason, it can be boldly declared For this reason, it can be boldly declared that an object moving in a circle at constant that an object moving in a circle at constant speed is indeed accelerating. speed is indeed accelerating.

It is accelerating because its velocity is It is accelerating because its velocity is changing its directions.changing its directions.

In which direction does the In which direction does the velocity change (Acceleration) velocity change (Acceleration)

vector point?vector point? The velocity change vector is The velocity change vector is

directed towards the center. directed towards the center. An object moving in a circle at a An object moving in a circle at a

constant speed from A to B constant speed from A to B experiences a velocity change and experiences a velocity change and therefore an acceleration.therefore an acceleration.

This acceleration is directed towards This acceleration is directed towards the center of the circlethe center of the circle..

Briefly explain why, using Briefly explain why, using geometric proof.geometric proof.

In this time, the velocity has changed from In this time, the velocity has changed from vvii to to vvff. The process . The process

of subtracting of subtracting vvii from from vvff is shown in the vector diagram; this is shown in the vector diagram; this

process yields the change in velocity.process yields the change in velocity. There is a velocity change for an object moving in a circle with a There is a velocity change for an object moving in a circle with a

constant speed. constant speed. Note that this velocity change vector is directed towards the Note that this velocity change vector is directed towards the

center. center.

•In the case of an object moving in a circle about point C, the object has moved from point A to point B.

Give two real world Give two real world demonstrations of this inward demonstrations of this inward

acceleration. acceleration. If a glass with a lit candle is held at the If a glass with a lit candle is held at the

end of an outstretched arm as you spin in end of an outstretched arm as you spin in a circle at a constant rate (such that the a circle at a constant rate (such that the flame experiences an acceleration), then flame experiences an acceleration), then the candle flame will no longer extend the candle flame will no longer extend vertically upwards. vertically upwards.

Instead the flame deflects from its upright Instead the flame deflects from its upright position, signifying that there is an position, signifying that there is an acceleration when the flame moves in a acceleration when the flame moves in a circular path at constant speed. circular path at constant speed.

The deflection of the flame will be in the The deflection of the flame will be in the direction of the acceleration. direction of the acceleration.

When you do this experiment, you find When you do this experiment, you find that the flame deflects towards the center that the flame deflects towards the center of the circle, thus indicating that not only of the circle, thus indicating that not only is there an acceleration; but that there is is there an acceleration; but that there is an inward acceleration.an inward acceleration.

Give two real world Give two real world demonstrations of this inward demonstrations of this inward

acceleration. acceleration. a.a. Use a cork accelerometer (a cork Use a cork accelerometer (a cork

submerged in a sealed flask of water) held submerged in a sealed flask of water) held in an outstretched arm and move in a in an outstretched arm and move in a circle at a constant rate of turning. circle at a constant rate of turning.

As the cork-water combination spun in the As the cork-water combination spun in the circle, the cork leaned towards the center circle, the cork leaned towards the center of the circle. of the circle.

Once more, there is proof that an object Once more, there is proof that an object moving in circular motion at constant moving in circular motion at constant speed experiences an acceleration which speed experiences an acceleration which directed towards the center of the circle.directed towards the center of the circle.

Do the Do the “C“Check Your heck Your

UnderstandingUnderstanding” ” #1-8 on the site.#1-8 on the site. Go to the website to use the Go to the website to use the

interactive questions.interactive questions.

An object moving in a circle is An object moving in a circle is experiencing acceleration. Even if experiencing acceleration. Even if moving around the perimeter of moving around the perimeter of the circle with a constant speed, the circle with a constant speed, there is still a change in velocity there is still a change in velocity

and subsequently an and subsequently an acceleration. acceleration.

This acceleration is directed This acceleration is directed TOWARDS THE CENTER of the TOWARDS THE CENTER of the

circle. circle.

Newton's second law of motionNewton's second law of motion says that … says that …

an object which experiences an an object which experiences an acceleration must also be acceleration must also be experiencing a net forceexperiencing a net force

the direction of the net force is the direction of the net force is in the same direction as the in the same direction as the acceleration. acceleration.

So for an object moving So for an object moving in a circle, there must be in a circle, there must be an inward force an inward force acting acting upon it in order to cause upon it in order to cause its inward acceleration.its inward acceleration.

This is sometimes This is sometimes referred to as the referred to as the

CENTRIPETAL FORCE CENTRIPETAL FORCE REQUIREMENT. REQUIREMENT.

The word "centripetal" The word "centripetal" means CENTER-SEEKING. means CENTER-SEEKING.

For objects moving in For objects moving in circular motion, there is a circular motion, there is a

net force acting towards the net force acting towards the center which causes the center which causes the object to object to seekseek the center. the center.

Newton's first law of motionNewton's first law of motion - - the law the law of inertia of inertia states that….states that….

"... objects in motion tend to stay "... objects in motion tend to stay in motion with the same speed in motion with the same speed and the same direction unless and the same direction unless acted upon by an unbalanced acted upon by an unbalanced force."force."

Objects will tend to naturally Objects will tend to naturally travel in straight lines; an travel in straight lines; an

unbalanced force is unbalanced force is required to cause it to turn. required to cause it to turn.

The presence of THE The presence of THE UNBALANCED FORCE is UNBALANCED FORCE is

required for objects to move required for objects to move in circles.in circles.

There is an outward force and an There is an outward force and an outward acceleration acting on your outward acceleration acting on your body when you sit in the passenger body when you sit in the passenger

seat of a car making a left hand turn. seat of a car making a left hand turn.

True or False?True or False?

Explain your answer. Explain your answer. It is the inertia of your body - the tendency It is the inertia of your body - the tendency

to resist acceleration - which causes it to to resist acceleration - which causes it to continue in its forward motion. continue in its forward motion.

While the car is accelerating inward, you While the car is accelerating inward, you continue in a straight line. continue in a straight line.

If you are sitting on the passenger side of If you are sitting on the passenger side of the car, then eventually the outside door of the car, then eventually the outside door of the car will hit you as the car turns inward. the car will hit you as the car turns inward.

In reality, you are continuing in your In reality, you are continuing in your straight-line inertial path tangent to the straight-line inertial path tangent to the circle while the car is accelerating out from circle while the car is accelerating out from under you. under you.

The sensation of an outward The sensation of an outward force and an outward force and an outward acceleration is a false sensation. acceleration is a false sensation.

There is no physical object There is no physical object capable of pushing you capable of pushing you outwards. outwards.

You are merely experiencing the You are merely experiencing the tendency of your body to tendency of your body to continue in its path tangent to continue in its path tangent to the circular path along which the the circular path along which the car is turning.car is turning.

What is a What is a centripetal forcecentripetal force? ?

Some physical force pushing or Some physical force pushing or pulling the object towards the pulling the object towards the center of the circle. center of the circle.

The word "centripetal" is merely The word "centripetal" is merely an adjective used to describe the an adjective used to describe the direction of the force. direction of the force.

Is centripetal force a new type Is centripetal force a new type of force? of force?

Yes or NoYes or No?? NO! NO! We are not introducing a new We are not introducing a new typetype of of

force but rather describing the force but rather describing the direction of the net force acting upon direction of the net force acting upon the object which moves in the circle. the object which moves in the circle.

Whatever the object, if it moves in a Whatever the object, if it moves in a circle, there is some force acting upon circle, there is some force acting upon it to cause it to deviate from its it to cause it to deviate from its straight-line path, accelerate inwards straight-line path, accelerate inwards and move along a circular path. and move along a circular path.

Give three examples of Give three examples of centripetal force. centripetal force.

As a car makes a turn, the force of As a car makes a turn, the force of friction acting upon the turned friction acting upon the turned wheels of the car provide the wheels of the car provide the

centripetal force required for circular centripetal force required for circular motion.motion.

Give three examples of Give three examples of centripetal force. centripetal force.

As a bucket of water is tied to a As a bucket of water is tied to a string and spun in a circle, the string and spun in a circle, the

force of tension acting upon the force of tension acting upon the bucket provides the centripetal bucket provides the centripetal

force required for circular force required for circular motion.motion.

Give three examples of Give three examples of centripetal force.centripetal force.

As the moon orbits the Earth, the As the moon orbits the Earth, the force of gravity acting upon the force of gravity acting upon the moon provides the centripetal moon provides the centripetal

force required for circular force required for circular motion.motion.

The centripetal force for uniform The centripetal force for uniform circular motion alters the circular motion alters the

direction of the object without direction of the object without altering its speed. altering its speed.

True or False?True or False?

Here’s the situation:Here’s the situation:

You are You are carrying a tennis carrying a tennis ball upon a flat, level board. ball upon a flat, level board.

Once the tennis ball and the Once the tennis ball and the board are in motion, they board are in motion, they will continue in motion in the will continue in motion in the same direction at the same same direction at the same speed unless acted upon by speed unless acted upon by an unbalanced force. an unbalanced force.

This is in accord with This is in accord with Newton's first law of motion. Newton's first law of motion.

Here’s the question:Here’s the question:

If you apply an unbalanced force If you apply an unbalanced force to the flat board, then the flat to the flat board, then the flat board will accelerate. board will accelerate.

If the force is continually directed If the force is continually directed towards a point at the center of towards a point at the center of the circle, then the flat board will the circle, then the flat board will round the corner in a circular-like round the corner in a circular-like path. path.

What will happen to the ball as What will happen to the ball as the flat board is moved in a the flat board is moved in a circle?circle?

The answer: The answer:

Go to the animation at this site. Go to the animation at this site. View the animation on the LEFT side View the animation on the LEFT side of the page.of the page.

http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/cf.htmlhttp://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/cf.html . .

An altered situation: An altered situation:

Now a block is secured to Now a block is secured to the board in such a manner the board in such a manner that the block applies an that the block applies an unbalanced force to the ball unbalanced force to the ball that is directed towards the that is directed towards the center of the circle.center of the circle.

The block provides a normal The block provides a normal force directed inward. force directed inward.

What will happen to the ball What will happen to the ball as the flat board is moved as the flat board is moved in a circle?in a circle?

The Answer: The Answer:

Go to the animation at this site. Go to the animation at this site. View the animation on the LEFT side View the animation on the LEFT side of the page. of the page.

http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/cf.htmlhttp://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/cf.html

Circular MotionCircular Motion

Centripetal: “towards the center”.Centripetal: “towards the center”.

Radial: Along the radius of a circle.Radial: Along the radius of a circle.

Centripetal is often used interchangeably Centripetal is often used interchangeably with radial.with radial.

Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

Moving around a Moving around a fixed point fixed point remaining the remaining the same distance same distance from that point at from that point at all times.all times.

Moving in a Moving in a straight line away straight line away from a fixed point, from a fixed point, increasing the increasing the distance from that distance from that point at all times.point at all times.

AXESAXES Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

Radial axis which Radial axis which points to the center points to the center of the circle.of the circle.

Tangential axis Tangential axis which points in the which points in the direction of motion.direction of motion.

Depending on Depending on point of view, a y-point of view, a y-axis may be used axis may be used as well.as well.

X-axis which X-axis which usually points usually points horizontally.horizontally.

Y-axis which Y-axis which usually points usually points vertically.vertically.

ACCELERATIONACCELERATION Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

Radial Acceleration Radial Acceleration always points to the always points to the center of the circle.center of the circle.

Tangential Tangential Acceleration Acceleration depends on depends on whether velocity is whether velocity is increasing, increasing, decreasing or decreasing or constant.constant.

X-axis acceleration X-axis acceleration depends on whether depends on whether velocity is increasing, velocity is increasing, decreasing or constant.decreasing or constant.

Y-axis acceleration Y-axis acceleration depends on whether depends on whether velocity is increasing, velocity is increasing, decreasing or constant.decreasing or constant.

Acceleration on one of Acceleration on one of the axes is always zero.the axes is always zero.

FORCESFORCES Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

There MUST be There MUST be at least one force at least one force pointing towards pointing towards the center of the the center of the circular path.circular path.

WHY?WHY?• Since there is always Since there is always

centripetal acceleration, centripetal acceleration, there must be a there must be a centripetal force. centripetal force.

• WHO said that?WHO said that? Newton (his second law)Newton (his second law)

There can be There can be forces in any forces in any direction.direction.

There must be a There must be a force pointing in force pointing in the direction of the direction of acceleration.acceleration.

IMPORTANT NOTEIMPORTANT NOTE

An object WILL NOT move in a An object WILL NOT move in a circular path unless there is a circular path unless there is a force acting on the radial axis force acting on the radial axis pointing in towards the center pointing in towards the center

of the circle.of the circle.

In which direction will the object In which direction will the object move without the centripetal force?move without the centripetal force? Along the tangential axisAlong the tangential axis In the direction of motionIn the direction of motion In a straight line.In a straight line.

MOTION DIAGRAMSMOTION DIAGRAMS Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

On the radial axis, On the radial axis, motion diagrams motion diagrams only consist of the only consist of the acceleration and acceleration and net force direction; net force direction; velocity is not velocity is not needed.needed.

On the tangential or On the tangential or y-axis, motion y-axis, motion diagrams are diagrams are normal, if needed.normal, if needed.

Draw velocity and Draw velocity and acceleration arrows acceleration arrows to represent to represent increasing, increasing, decreasing or decreasing or constant speed.constant speed.

FREE BODY DIAGRAMSFREE BODY DIAGRAMSCircular MotionCircular Motion vs. vs. Linear MotionLinear Motion

Draw forces the same Draw forces the same as always.as always.

Make sure there is a Make sure there is a force on the radial force on the radial axis pointing towards axis pointing towards the center of the the center of the circle.circle.

Axes must include the Axes must include the radial (positive radial (positive towards the center) towards the center) and either tangential and either tangential or y-axis.or y-axis.

Forces are Forces are represented with represented with arrows pointing in arrows pointing in the direction of the the direction of the push or pull.push or pull.

X- and y-axes point X- and y-axes point in the direction of in the direction of the most forces, the most forces, usually with x in the usually with x in the direction of motion.direction of motion.

EquationsEquations Circular MotionCircular Motion vs. vs. Linear MotionLinear Motion

Newton’s Newton’s Second Law: Second Law:

F = maF = ma Acceleration: Acceleration:

a = va = v22/R/R

Newton’s Newton’s Second Law: Second Law:

F = maF = ma Acceleration: Acceleration:

a = (va = (vff – v – v

oo)/t)/t

(You will recognize this (You will recognize this equation as vequation as vff = v = v00 + at) + at)

WHAT IS A CENTRIFUGAL WHAT IS A CENTRIFUGAL FORCE?FORCE?

Centrifugal forces are a MYTH! They Centrifugal forces are a MYTH! They do not really exist.do not really exist.

It is the effect of Newton’s First Law It is the effect of Newton’s First Law acting on an object while the object acting on an object while the object is being moved in a circular path.is being moved in a circular path.

What are three mathematical What are three mathematical quantities which will be of primary quantities which will be of primary

interest to us?interest to us?

force, speed, and acceleration.force, speed, and acceleration.

Velocity is determined using Velocity is determined using the equationthe equation

The acceleration of an object The acceleration of an object moving in a circle can be moving in a circle can be

determined by the equationsdetermined by the equations

The net force is related to the The net force is related to the acceleration of the object (as is acceleration of the object (as is always the case) and is given by always the case) and is given by

the equationsthe equations

This set of circular motion This set of circular motion equations can be used in two equations can be used in two

ways:ways:a.a. as a as a recipe for algebraic problem recipe for algebraic problem

solving solving in order to solve for an in order to solve for an unknown quantity. unknown quantity.

b.b. as a "guide to thinking" about as a "guide to thinking" about how how an alteration in one quantity an alteration in one quantity would effect a second quantitywould effect a second quantity

..

How can the equations for How can the equations for circular motion be used as a circular motion be used as a

guide to thinking? guide to thinking? the equation relating the net force the equation relating the net force

((FFnetnet) to the speed () to the speed (vv) of an object ) of an object moving in uniform circular motion is moving in uniform circular motion is FFnetnet = mv = mv22/R/R

This equation shows that the net This equation shows that the net force required for an object to move force required for an object to move in a circle is directly proportional to in a circle is directly proportional to the square of the speed of the object. the square of the speed of the object. For a constant mass and radius, the For a constant mass and radius, the FFnetnet is proportional to the is proportional to the speedspeed22..

The factor by which the net force is The factor by which the net force is altered is the square of the factor altered is the square of the factor by which the speed is altered. by which the speed is altered. Subsequently, if the speed of the Subsequently, if the speed of the object is doubled, the net force object is doubled, the net force required for that object's circular required for that object's circular motion is quadrupled. And if the motion is quadrupled. And if the speed of the object is halved speed of the object is halved (decreased by a factor of 2), the net (decreased by a factor of 2), the net force required is decreased by a force required is decreased by a factor of 4.factor of 4.

The process of solving a circular The process of solving a circular motion problem is much like any other motion problem is much like any other problem in physics class. problem in physics class.

The process involves:The process involves:• a careful reading of the problema careful reading of the problem• the identification of the known and the identification of the known and

required information in variable formrequired information in variable form• the selection of the relevant equation(s)the selection of the relevant equation(s)• substitution of known values into the substitution of known values into the

equationequation• finally algebraic manipulation of the finally algebraic manipulation of the

equation to determine the answer equation to determine the answer

A 900-kg car moving at 10 m/s A 900-kg car moving at 10 m/s takes a turn around a circle with a takes a turn around a circle with a radius of 25.0 m. Determine the radius of 25.0 m. Determine the acceleration and the net force acceleration and the net force

acting upon the car.acting upon the car. To determine the acceleration of the car, use To determine the acceleration of the car, use the equation a = (vthe equation a = (v22)/R. The solution is as )/R. The solution is as follows:follows:

a = (va = (v22)/R a = ((10.0 m/s))/R a = ((10.0 m/s)22)/(25.0 m))/(25.0 m)

a = (100 ma = (100 m22/s/s22)/(25.0 m))/(25.0 m)

a = 4 m/sa = 4 m/s22

To determine the net force acting upon the car, To determine the net force acting upon the car, use the equation Fnet = m*a. The solution is as use the equation Fnet = m*a. The solution is as follows.follows.

FFnetnet = m*a F = m*a Fnetnet = (900 kg)*(4 m/s = (900 kg)*(4 m/s22))

FFnetnet = 3600 N = 3600 N

A 95-kg halfback makes a turn on the football field. A 95-kg halfback makes a turn on the football field. The halfback sweeps out a path which is a portion of The halfback sweeps out a path which is a portion of a circle with a radius of 12-meters. The halfback a circle with a radius of 12-meters. The halfback makes a quarter of a turn around the circle in 2.1 makes a quarter of a turn around the circle in 2.1

seconds. Determine the speed of the halfback.seconds. Determine the speed of the halfback. To determine the speed of the To determine the speed of the

halfback, use the equation v = d/t halfback, use the equation v = d/t where the d is one-fourth of the where the d is one-fourth of the circumference and the time is 2.1 s. circumference and the time is 2.1 s. The solution is as follows:The solution is as follows:

v = d/t v = (0.25 * 2 * pi * R)/tv = d/t v = (0.25 * 2 * pi * R)/t

v = (0.25 * 2 * 3.14 * 12.0 m)/(2.1 s)v = (0.25 * 2 * 3.14 * 12.0 m)/(2.1 s)

v = 8.97 m/sv = 8.97 m/s

A 95-kg halfback makes a turn on the football field. A 95-kg halfback makes a turn on the football field. The halfback sweeps out a path which is a portion The halfback sweeps out a path which is a portion of a circle with a radius of 12-meters. The halfback of a circle with a radius of 12-meters. The halfback makes a quarter of a turn around the circle in 2.1 makes a quarter of a turn around the circle in 2.1 seconds. Determine the acceleration of and net seconds. Determine the acceleration of and net

force acting upon the halfback.force acting upon the halfback. To determine the acceleration of the halfback, use To determine the acceleration of the halfback, use

the equation a = (vthe equation a = (v22)/R. The solution is as follows:)/R. The solution is as follows:

a = (va = (v22)/R a = ((8.97 m/s))/R a = ((8.97 m/s)22)/(12.0 m))/(12.0 m)

a = (80.5 ma = (80.5 m22/s/s22)/(12.0 m))/(12.0 m)

a = 6.71 m/sa = 6.71 m/s22

To determine the net force acting upon the To determine the net force acting upon the halfback, use the equation Fnet = m*a. The halfback, use the equation Fnet = m*a. The solution is as follows.solution is as follows.

FFnetnet = m*a F = m*a Fnetnet = (95.0 kg)*(6.71 m/s = (95.0 kg)*(6.71 m/s22))

FFnetnet = 637 = 637 N N

Anna fills a bucket with water, ties it to a Anna fills a bucket with water, ties it to a strong rope, and spins it in a circle. Anna strong rope, and spins it in a circle. Anna

spins the bucket when it is half-full of water spins the bucket when it is half-full of water and when it is quarter-full of water. In which and when it is quarter-full of water. In which

case is more force required to spin the case is more force required to spin the bucket in a circle? bucket in a circle?

Answer the question and then check Answer the question and then check at The Physics Classroom!at The Physics Classroom!

A Lincoln Continental and a Yugo are making A Lincoln Continental and a Yugo are making a turn. The Lincoln is four times more massive a turn. The Lincoln is four times more massive

than the Yugo. If they make the turn at the than the Yugo. If they make the turn at the same speed, then how do the centripetal same speed, then how do the centripetal forces acting upon the two cars compare. forces acting upon the two cars compare.

Explain.Explain.


The Gravitron at Six Flags is a ride in which The Gravitron at Six Flags is a ride in which occupants line the perimeter of a cylinder occupants line the perimeter of a cylinder

and spin in a circle at a high rate of turning. and spin in a circle at a high rate of turning. When the cylinder begins spinning very When the cylinder begins spinning very

rapidly, the floor is removed from under the rapidly, the floor is removed from under the riders' feet. What effect does a doubling in riders' feet. What effect does a doubling in

speed have upon the centripetal force? speed have upon the centripetal force? Explain.Explain.


Determine the centripetal force acting upon Determine the centripetal force acting upon a 40-kg child who makes 10 revolutions a 40-kg child who makes 10 revolutions

around the Cliffhanger in 29.3 seconds. The around the Cliffhanger in 29.3 seconds. The radius of the barrel is 2.90 meters.radius of the barrel is 2.90 meters.


Technology

Circular Motion PPP