Dynamics of Uniform Circular Motion

Chapter 5

Learning Objectives- Circular motion and rotation Uniform circular motion Students should understand the uniform circular motion of a particle, so they

can: Relate the radius of the circle and the speed or rate of revolution of the

particle to the magnitude of the centripetal acceleration. Describe the direction of the particle’s velocity and acceleration at any

instant during the motion. Determine the components of the velocity and acceleration vectors at

any instant, and sketch or identify graphs of these quantities. Analyze situations in which an object moves with specified

acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or of one of the forces that makes up the net force, in situations such as the following:• Motion in a horizontal circle (e.g., mass on a rotating merry-go-

round, or car rounding a banked curve).• Motion in a vertical circle (e.g., mass swinging on the end of a

string, cart rolling down a curved track, rider on a Ferris wheel).

Table Of Contents

5.1 Uniform Circular Motion

5.2 Centripetal Acceleration

5.3 Centripetal Force

5.4 Banked Curves

5.5 Satellites in Circular Orbits

5.6 Apparent Weightlessness and Artificial Gravity

5.7 Vertical Circular Motion

Chapter 5:

Dynamics of Uniform Circular Motion

Section 1:

Uniform Circular Motion

Other Effects of Forces

Up until now, we’ve focused on forces that speed up

or slow down an object.

We will now look at the third effect of a force

Turning

We need some other equations as the object will be

accelerating without necessarily changing speed.

Uniform circular motion is the motion of an object traveling at a constant speed on a circular path.

DEFINITION OF UNIFORM CIRCULAR MOTION

Let T be the time it takes for the object totravel once around the circle.

vr

T

2

r

Example 1: A Tire-Balancing Machine

The wheel of a car has a radius of 0.29m and it being rotatedat 830 revolutions per minute on a tire-balancing machine. Determine the speed at which the outer edge of the wheel ismoving.

revolutionmin102.1minsrevolution830

1 3

s 072.0min 102.1 3 T

sm25

s 072.0

m 0.2922

T

rv

Newton’s Laws

1st

When objects move along a straight line the sideways/perpendicular forces must be balanced.

2nd When the forces directed perpendicular to velocity become

unbalanced the object will curve. 3rd

The force that pulls inward on the object, causing it to curve off line provides the action force that is centripetal in nature. The object will in return create a reaction force that is centrifugal in nature.

5.1.1. An airplane flying at 115 m/s due east makes a gradual turn while maintaining its speed and follows a circular path to fly south. The turn takes 15 seconds to complete. What is the radius of the circular path?

a) 410 m

b) 830 m

c) 1100 m

d) 1600 m

e) 2200 m

Chapter 5:

Dynamics of Uniform Circular Motion

Section 2:

Centripetal Acceleration

In uniform circular motion, the speed is constant, but the direction of the velocity vector is not constant.

9090

r

tv

v

v

r

v

t

v 2

r

vac

2

The direction of the centripetal acceleration is towards the center of the circle; in the same direction as the change in velocity.

r

vac

2

Conceptual Example 2: Which Way Will the Object Go?

An object is in uniform circular motion. At point O it is released from its circular path. Does the object move along the straightpath between O and A or along the circular arc between pointsO and P ?

Straight path

Example 3: The Effect of Radius on Centripetal Acceleration

The bobsled track contains turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s. Express answers as multiples of .sm8.9 2g

rvac2

m 33

sm34 2

ca

m 24

sm34 2

ca

2sm35 g6.3

2sm48 g9.4

5.2.1. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two?

a) Keep the radius fixed and increase the period by a factor of two.

b) Keep the radius fixed and decrease the period by a factor of two.

c) Keep the speed fixed and increase the radius by a factor of two.

d) Keep the speed fixed and decrease the radius by a factor of two.

e) Keep the radius fixed and increase the speed by a factor of two.

5.2.2. A steel ball is whirled on the end of a chain in a horizontal circle of radius R with a constant period T. If the radius of the circle is then reduced to 0.75R, while the period remains T, what happens to the centripetal acceleration of the ball?

a) The centripetal acceleration increases to 1.33 times its initial value.

b) The centripetal acceleration increases to 1.78 times its initial value.

c) The centripetal acceleration decreases to 0.75 times its initial value.

d) The centripetal acceleration decreases to 0.56 times its initial value.

e) The centripetal acceleration does not change.

5.2.3. While we are in this classroom, the Earth is orbiting the Sun in an orbit that is nearly circular with an average radius of 1.50 × 1011 m. Assuming that the Earth is in uniform circular motion, what is the centripetal acceleration of the Earth in its orbit around the Sun?

a) 5.9 × 103 m/s2

b) 1.9 × 105 m/s2

c) 3.2 × 107 m/s2

d) 7.0 × 102 m/s2

e) 9.8 m/s2

5.2.4. A truck is traveling with a constant speed of 15 m/s. When the truck follows a curve in the road, its centripetal acceleration is 4.0 m/s2. What is the radius of the curve?

a) 3.8 m

b) 14 m

c) 56 m

d) 120 m

e) 210 m

5.2.5. Consider the following situations:(i) A minivan is following a hairpin turn on a mountain road at a

constant speed of twenty miles per hour.(ii) A parachutist is descending at a constant speed 10 m/s.(iii) A heavy crate has been given a quick shove and is now sliding

across the floor.(iv) Jenny is swinging back and forth on a swing at the park.(v) A football that was kicked is flying through the goal posts.(vi) A plucked guitar string vibrates at a constant frequency.In which one of these situations does the object or person

experience zero acceleration?

a) i only b) ii only

c) iii and iv only d) iv, v, and vi only

e) all of the situations

Chapter 5:

Dynamics of Uniform Circular Motion

Section 3:

Centripetal Force

Recall Newton’s Second Law

aF

mm

F

a

When a net external force acts on an objectof mass m, the acceleration that results is directly proportional to the net force and hasa magnitude that is inversely proportional tothe mass. The direction of the acceleration isthe same as the direction of the net force.

r

vmmaF cc

2

Recall Newton’s Second Law

Thus, in uniform circular motion there must be a net force to produce the centripetal acceleration.

The centripetal force is the name given to the net force required to keep an object moving on a circular path.

The direction of the centripetal force always points toward the center of the circle and continually changes direction as the object moves.

Problem Solving Strategy – Horizontal Circles

1. Draw a free-body diagram of the curving object(s).

2. Choose a coordinate system with the following two axes.

a) One axis will point inward along the radius (inward is positive direction).

b) One axis will point perpendicular to the circular path (up is positive direction).

3. Sum the forces along each axis to get two equations for two unknowns.

a) FRADIUS: +FIN FOUT = m(v2)/ r

b) F : FUP FDOWN = 0

4. Do the math of two equations with two unknowns.

R

Just in case…

The third dimension in these problems would be a direction tangent to the circle and in the plane of the circle.

We choose to ignore this direction for objects moving at constant speed.

If an object moves along the circle with changing speed then the forces tangent to the circle have become unbalanced.

You can sum the tangential forces to find the rate at which speed changes with time, aTAN.

The linear kinematics equations can then be used to describe motion along or tangent to the circle. FTAN: FFORWARD FBACKWARD = m aTAN

R

tan

Example 5: The Effect of Speed on Centripetal Force

The model airplane has a mass of 0.90 kg and moves at constant speed on a circle that is parallel to the ground. The path of the airplane and the guideline lie in the samehorizontal plane because the weight of the plane is balancedby the lift generated by its wings. Find the tension in the 17 mguideline for a speed of 19 m/s.

r

vmTFc

2

m 17

sm19kg 90.0

2

T N 19

5.3.1. A boy is whirling a stone at the end of a string around his head. The string makes one complete revolution every second, and the tension in the string is FT. The boy increases the speed of the stone, keeping the radius of the circle unchanged, so that the string makes two complete revolutions per second. What happens to the tension in the sting?

a) The tension increases to four times its original value.

b) The tension increases to twice its original value.

c) The tension is unchanged.

d) The tension is reduced to one half of its original value.

e) The tension is reduced to one fourth of its original value.

5.3.2. An aluminum rod is designed to break when it is under a tension of 600 N. One end of the rod is connected to a motor and a 12-kg spherical object is attached to the other end. When the motor is turned on, the object moves in a horizontal circle with a radius of 6.0 m. If the speed of the motor is continuously increased, at what speed will the rod break? Ignore the mass of the rod for this calculation.

a) 11 m/s

b) 17 m/s

c) 34 m/s

d) 88 m/s

e) 3.0 × 102 m/s

5.3.3. A ball is attached to a string and whirled in a horizontal circle. The ball is moving in uniform circular motion when the string separates from the ball (the knot wasn’t very tight). Which one of the following statements best describes the subsequent motion of the ball?

a) The ball immediately flies in the direction radially outward from the center of the circular path the ball had been following.

b) The ball continues to follow the circular path for a short time, but then it gradually falls away.

c) The ball gradually curves away from the circular path it had been following.

d) The ball immediately follows a linear path away from, but not tangent to the circular path it had been following.

e) The ball immediately follows a line that is tangent to the circular path the ball had been following

5.3.4. A rancher puts a hay bail into the back of her SUV. Later, she drives around an unbanked curve with a radius of 48 m at a speed of 16 m/s. What is the minimum coefficient of static friction for the hay bail on the floor of the SUV so that the hay bail does not slide while on the curve?

a) This cannot be determined without knowing the mass of the hay bail.

b) 0.17

c) 0.33

d) 0.42

e) 0.54

5.3.5. Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground (a horizontal circle). What is the force, the physical force, holding the bucket in a circular path?

a) the centripetal force

b) the centrifugal force

c) your hand on the handle

d) gravitational force

e) None of the above are correct.

5.3.6. Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground (a horizontal circle). Now imagine there's a ball in the bucket. What keeps the ball moving in a circular path?

a) contact force of the bucket on the ball

b) contact force of the ball on the bucket

c) gravitational force on the ball

d) the centripetal force

e) the centrifugal force

5.3.7. The moon, which is approximately 4 × 109 m from Earth, has a mass of 7.4 × 1022 kg and a period of 27.3 days. What must is the magnitude of the gravitational force between the Earth and the moon?

a) 1.8 × 1018 N

b) 2.1 × 1022 N

c) 1.7 × 1013 N

d) 5.0 × 1022 N

e) 4.2 × 1020 N

5.3.8. The Rapid Rotor amusement ride is spinning fast enough that the floor beneath the rider drops away and the rider remains in place. If the Rotor speeds up until it is going twice as fast as it was previously, what is the effect on the frictional force on the rider?

a) The frictional force is reduced to one-fourth of its previous value.

b) The frictional force is the same as its previous value.

c) The frictional force is reduced to one-half of its previous value.

d) The frictional force is increased to twice its previous value.

e) The frictional force is increased to four times its previous value.

Chapter 5:

Dynamics of Uniform Circular Motion

Section 4:

Banked Curves

On an unbanked curve, the static frictional forceprovides the centripetal force.

Unbanked curve

On a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight.

Banked Curve

r

vmFF Nc

2

sin

mgFN cosrg

v2

tan

Example 8: The Daytona 500

The turns at the Daytona International Speedway have a maximum radius of 316 m and are steely banked at 31degrees. Suppose these turns were frictionless. As what speed would the cars have to travel around them?

rg

v2

tan tanrgv

31tansm8.9m 316 2v

mph 96 sm43v

5.4.1. Complete the following statement: The maximum speed at which a car can safely negotiate an unbanked curve depends on all of the following factors except

a) the coefficient of kinetic friction between the road and the tires.

b) the coefficient of static friction between the road and the tires.

c) the acceleration due to gravity.

d) the diameter of the curve.

e) the ratio of the static frictional force between the road and the tires and the normal force exerted on the car.

5.4.2. A 1000-kg car travels along a straight portion of highway at a constant velocity of 10 m/s, due east. The car then encounters an unbanked curve of radius 50 m. The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from east to south. What is the magnitude of the acceleration of the car as it travels the unbanked curve?

a) zero m/s2

b) 2 m/s2

c) 5 m/s2

d) 10 m/s2

e) 20 m/s2

5.4.3. A 1000-kg car travels along a straight portion of highway at a constant velocity of 10 m/s, due east. The car then encounters an unbanked curve of radius 50 m. The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from east to south. What is the magnitude of the frictional force between the tires and the road as the car negotiates the unbanked curve?

a) 500 N

b) 1000 N

c) 2000 N

d) 5000 N

e) 10 000 N

5.4.4. You are riding in the forward passenger seat of a car as it travels along a straight portion of highway. The car continues traveling at a constant speed as it follows a sharp, unbanked curve to the left. You feel the door pushing on the right side of your body. Which of the following forces in the horizontal direction are acting on you?

a) a static frictional force between you and the seat

b) a normal force of the door

c) a force pushing you toward the door

d) answers a and b

e) answers a and c

Chapter 5:

Dynamics of Uniform Circular Motion

Section 5:

Satellites in Circular Orbits

There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.

Don’t worry, it’s only rocket science

2r

mMG E

r

GMv E

r

vmFc

2

Example 9: Orbital Speed of the Hubble Space Telescope

Determine the speed of the Hubble Space Telescope orbitingat a height of 598 km above the earth’s surface.

m10598m1038.6

kg1098.5kgmN1067.636

242211

v

mph 6,9001 sm1056.7 3 v

22MACHv

r

GMv E

T

r

r

GMv E 2

EGM

rT

232

Period to orbit the Earth

hours 24T

EGM

rT

232

Geosynchronous Orbit

3

2

243

112

2

109742.51067.6400,86

kgxskg

mxs

r

3

2

2

2EGMT

r 3

2

2

2EGMT

r

mxr 71022.4 mxrE61038.6

milesmxh 000,221058.3 7

5.5.1. A satellite is in a circular orbit around the Earth. If it is at an altitude equal to twice the radius of the Earth, 2RE, how does its speed v relate to the Earth's radius RE, and the magnitude g of the acceleration due to gravity on the Earth's surface?

a) b)

c) d)

e)

EgRv31 EgRv

21

EgRv 2 EgRv21

EgRv31

5.5.2. It is the year 2094; and people are designing a new space station that will be placed in a circular orbit around the Sun. The orbital period of the station will be 6.0 years. Determine the ratio of the station’s orbital radius about the Sun to that of the Earth’s orbital radius about the Sun. Assume that the Earth’s obit about the Sun is circular.

a) 2.4

b) 3.3

c) 4.0

d) 5.2

e) 6.0

5.5.3. A space probe is orbiting a planet on a circular orbit of radius R and a speed v. The acceleration of the probe is a. Suppose rockets on the probe are fired causing the probe to move to another circular orbit of radius 0.5R and speed 2v. What is the magnitude of the probe’s acceleration in the new orbit?

a) a/2

b) a

c) 2a

d) 4a

e) 8a

Chapter 5:

Dynamics of Uniform Circular Motion

Section 6:

Apparent Weightlessness and Artificial Gravity

Conceptual Example 12: Apparent Weightlessness and Free Fall

In each case, what is the weight recorded by the scale?

Example 13: Artificial Gravity

At what speed must the surface of the space station moveso that the astronaut experiences a push on his feet equal to his weight on earth? The radius is 1700 m.

mgr

vmFc

2

2sm80.9m 1700

rgv

smv 130

5.6.1. A space station is designed in the shape of a large, hollow donut that is uniformly rotating. The outer radius of the station is 460 m. With what period must the station rotate so that a person sitting on the outer wall experiences “artificial gravity,” i.e. an acceleration of 9.8 m/s2?

a) 43 s

b) 76 s

c) 88 s

d) 110 s

e) 230 s

Chapter 5:

Dynamics of Uniform Circular Motion

Section 7:

Vertical Circular Motion

Circular Motion

In the previous lesson the radial and the perpendicular forces were emphasized while the tangential forces were ignored. Each class of forces serves a different function for objects moving along a circle.

Class of Force Purpose of the Force

Radial Forces Curves the object off a straight-line path.

Perpendicular Forces Holds the object in the plane of the circle.

Tangential Forces Changes the speed of the object along the circle.

Circular Motion

Most of the horizontal, circular problems occurred at constant speed so that we could ignore the tangential forces. The vertical, circular problems have objects moving with and against gravity so that speed changes. Tangential forces become significant. The good news is that perpendicular forces can now be ignored unless hurricanes are present.

Problem Solving Strategy for Vertical Circles

1. Draw a free-body diagram for the curving objects.

2. Choose a coordinate system with the following two axes.

a) One axis will point inward along the radius.

b) One axis points tangent to the circle in the circular plane, along the direction

of motion.

3. Sum the forces along each axis to get two equations for two unknowns.

a) FRADIUS: +FIN FOUT = m(v2)/ r b) FTAN : FFORWARD FBACKWARDS = ma

4. You can generally expect the weight of the object to have components in both

equations unless the object is exactly at the top, bottom or sides of the circle.

5. If the object changes height along the circle you may need to write a

conservation of energy statement. This goes well with centripetal forces since

there is an {mv2} in both kinetic energy terms and in centripetal force terms.

6. Do the math with 3(a) and 4 or perhaps 3(a) and 3(b).

Minimum/Maximum Speed Problems

Sometimes the problem addresses “the minimum speed” that an object can move through the top of the circle or “maximum speed” that an object can move along the top of the circle.

If the bucket of water turns too slowly you get wet. If a car tops a hill too quickly it leaves the ground. Allowing v2/r to equal g can solve many of these questions. By solving for v you will find a critical speed.

WC FF mgr

mv

2

rgv

Conceptual Example: A Trapeze Act

In a circus, a man hangs upside down from a trapeze, legsbent over and arms downward, holding his partner. Is it harderfor the man to hold his partner when the partner hangs straight down and is stationary of when the partner is swingingthrough the straight-down position?

r

vmmgFN

21

1

r

vmmgFN

23

3

r

vmFN

22

2

r

vmFN

24

4

5.7.1. At a circus, a clown on a motorcycle with a mass M travels along a horizontal track and enters a vertical circle of radius r. Which one of the following expressions determines the minimum speed that the motorcycle must have at the top of the track to remain in contact with the track?

a) b)

c) v = gR d) v = 2gR

e) v = MgR

grv 2 grv

5.7.2. A ball on the end of a rope is moving in a vertical circle near the surface of the earth. Point A is at the top of the circle; C is at the bottom. Points B and D are exactly halfway between A and C. Which one of the following statements concerning the tension in the rope is true?

a) The tension is the same at points A and C.

b) The tension is smallest at point C.

c) The tension is smallest at both points B and D.

d) The tension is smallest at point A.

e) The tension is the same at all four points.

5.7.3. An aluminum rod is designed to break when it is under a tension of 650 N. One end of the rod is connected to a motor and a 12-kg spherical object is attached to the other end. When the motor is turned on, the object moves in a vertical circle with a radius of 6.0 m. If the speed of the motor is continuously increased, what is the maximum speed the object can have at the bottom of the circle without breaking the rod? Ignore the mass of the rod for this calculation.

a) 4.0 m/s

b) 11 m/s

c) 16 m/s

d) 128 m/s

e) 266 m/s

5.7.4. A girl is swinging on a swing in the park. As she wings back and forth, she follows a path that is part of a vertical circle. Her speed is maximum at the lowest point on the circle and temporarily zero m/s at the two highest points of the motion as her direction changes. Which of the following forces act on the girl when she is at the lowest point on the circle?

a) the force of gravity, which is directed downward

b) the force which is directed radially outward from the center of the circle

c) the tension in the chains of the swing, which is directed upward

d) answers b and c only

e) answers a and c only

5.7.5 Which of the following parameters determine how fast you need to swing a water bucket vertically so that water in the bucket will not fall out?

a) radius of swing

b) mass of bucket

c) mass of water

d) a and b

e) a and c