Centripetal Force Acting on the Mass?

5/13/14

Objectives Centripetal • Describe and analyze the motion of force objects moving in circular motion. • Apply Newton’s second law to circular motion problems.

• Interpret free-body force diagrams.

Assessment Assessment

1. A race car is moving with a speed of 200 km/h on a circular section 2. In the case of a car rounding a turn, what real physical of a race track that has a radius of 300 m. The race car and driver force is acting radially to keep the car moving in a have a mass of 800 kg. circular path?

a. What is the magnitude of the centripetal acceleration felt by the driver?

b. What is the centripetal force acting on the mass?

Physics terms Equations

• centripetal force

• centripetal acceleration

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Consider Newton’s first law Circular motion

An object in motion remains in motion in a straight line at Imagine whirling a yo-yo constant speed unless acted upon by an unbalanced force. around your head in a circle.

Do you have to exert a force on the yo-yo?

Circular motion Investigation

Imagine whirling a yo-yo around your head in a circle. In Investigation 7A you will examine the velocity, Do you have to exert a force acceleration, and force on the yo-yo? vectors for an object in circular motion. Yes!

Click this Which way does this force simulation vector point? on page 210.

Investigation Investigation

Part 1: Directions of the vectors Questions for Part 1

1. Set m = 5.0 kg, r = 5.0 m, and a. Which vector quantities are v = 5.0 m/s. radial and which are tangential? Are the radial quantities pointed 2. Play the simulation and into the center or outwards? pause it at various positions around the circle. b. Do the lengths of the velocity, acceleration, or force vectors 3. Sketch the velocity, force, change around the circle? and acceleration vectors for at least five positions c. Notice that the angular velocity distributed around the circle. is exactly 1 rad/sec. Why?

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Investigation Investigation

Part 2: Approximating a mass swung overhead Questions for Part 2

1. Set r = 1.0 m and m = 0.3 kg a. How much force is needed to maintain the object in circular 2. Calculate the tangential motion? velocity needed to spin the object once per second, and b. Compare this with the force enter into simulation. required to hold the object motionless against the force of gravity.

c. Lengthen the string to r = 2.0 m. Do you need more or less force to maintain the same angular velocity?

Investigation Investigation

Part 3: Variation of velocity with radius Questions for Part 3

1. Hold the force constant at 10 N a. Graph v (on the vertical axis) and the mass constant at 2 kg, against r and describe the but vary the length of the string shape of your graph. from r = 1 m to 5 m. b. Graph v2 against r, describe the 2. Record the velocity and radius shape of your graph, and for each case. measure its slope (including units).

Is it accelerating? Acceleration at constant speed

v This object is moving at constant This object is moving at constant speed. Is it accelerating? speed. Is it accelerating?

Hint: YES! It is accelerating. The Is the velocity vector changing? velocity vector is constantly v changing direction.

v v

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Acceleration at constant speed Centripetal acceleration

v This object is moving at constant Centripetal acceleration . . . speed. Is it accelerating? • constantly redirects the velocity vector. v • always points toward the center of the circle.

Uniform circular motion is the weird case of acceleration at v ac v ac constant speed. ac ac

v v

Centripetal acceleration Centripetal acceleration

How can you calculate ac? How can you calculate ac?

What is the mathematical What is the mathematical relationship between velocity relationship between velocity and centripetal acceleration? and centripetal acceleration?

v ac v ac a a c Where does this c equation come from? v v

Centripetal acceleration Centripetal acceleration

Derivation: Centripetal acceleration is:

An object in circular motion moves from point A to point B at constant speed. • The distance from A to B is d = vΔt. The blue and gray triangles are similar. • The change in velocity is Δv. Use this to get an expression for Δv.

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Centripetal acceleration Centripetal acceleration

Similar triangles mean that: Similar triangles mean that:

Therefore, centripetal acceleration is: Therefore, centripetal acceleration is:

Notice: Δv always points toward the center of the circle.

Exploring the ideas Engaging with the concepts

What is ac for a mass spun around on a 1.0-m-long string if the mass completes one Click the rotation every second? interactive calculator on Hint: First, find the velocity. page 211. Centripetal acceleration 1.0

Engaging with the concepts Engaging with the concepts

Click to view motion

What is ac for a mass spun What is ac for a mass spun around on a 1.0-m-long string around on a 1.0-m-long string if the mass completes one 6.28 if the mass completes one 6.28 rotation every second? rotation every second?

39.5

Centripetal acceleration Centripetal acceleration 1.0 1.0

Now find the acceleration. Now find the acceleration.

This is about 4 g’s of acceleration!

It’s not unusual for ac to be large.

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Centripetal force Centripetal force

It takes a force to cause an Any type of physical force can object to move in a circle. provide a centripetal force.

This force is called the What supplies the centripetal centripetal force. force in this situation?

Centripetal force Centripetal force

Fc = T Fc = T Any type of physical force can The centripetal force is the provide a centripetal force. force or combination of forces that point toward the center of What supplies the centripetal the circle. force in this situation?

The string supplies the centripetal force.

The centripetal force is the T T force of tension.

Centripetal force Centripetal force

What if the string breaks? What if the string breaks? Which way will the yo-yo go? Which way will the yo-yo go?

It flies off in a straight line tangent to the circle.

It can’t stay in the circle unless a centripetal force is applied.

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Newton’s second law Centripetal force

A centripetal force causes an Combining these two equations . . . object to undergo centripetal acceleration. and

The centripetal force and acceleration vectors must point Fc in the SAME direction: toward ac the center of the circle.

Centripetal force Exploring the ideas

Combining these two equations . . .

and Click the

interactive

calculator on you get: page 211.

Engaging with the concepts Engaging with the concepts

A 200 gram yo-yo on a 1.0 meter A 200 gram yo-yo on a 1.0 meter string is spinning in a circle string is spinning in a circle once every second. What is the 0.200 once every second. What is the 0.200 6.28 tension in the string? tension in the string?

Hint: First find the velocity. Centripetal force Centripetal force 1.0 1.0

Next, find the centripetal force.

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Engaging with the concepts Engaging with the concepts

A 200 gram yo-yo on a 1.0 meter A 1000 kg car traveling at 10 m/s string is spinning in a circle is rounding a curve with a radius once every second. What is the 0.200 6.28 of 50 m. How much force is 1000 10 tension in the string? needed to keep it on the road?

7.9 Centripetal force Centripetal force 1.0 50

Engaging with the concepts Engaging with the concepts

A 1000 kg car traveling at 10 m/s A 1000 kg car traveling at 10 m/s is rounding a curve with a radius is rounding a curve with a radius of 50 m. How much force is 1000 10 of 50 m. How much force is 1000 20 needed to keep it on the road? needed to keep it on the road?

2000 8000 Centripetal force Centripetal force 50 50

If the speed doubles, how much more force is needed? If the speed doubles, how much more force is needed? four times more (8000 N)

Centrifugal vs. centripetal force Centrifugal force

The terms centripetal and centrifugal sound so similar. What’s the Imagine that you are a small child in a difference? car making a sharp turn.

• Centripetal means “center-seeking”. As the car turns, you slide sideways A centripetal force pushes inward, toward the center of the circle. across the seat until you are pressed against the far wall of the car (unless • Centrifugal means “center-fleeing”. you wear a seatbelt! ). A centrifugal force pushes outward, away from the circle. It FEELS like a force is pushing you One of these forces is referred to as “fictitious”. OUT of the circle. Which one, and why? Why?

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Centrifugal force Centrifugal force

It FEELS like a force is pushing you It FEELS like a force is pushing you OUT of the circle. OUT of the circle.

Why? Why?

You are obeying Newton’s first law. You are obeying Newton’s first law. You are moving in a straight line You are moving in a straight line UNLESS acted upon by a force. UNLESS acted upon by a force.

There is NO ACTUAL FORCE pushing you out of the circle. That is why say the centrifugal force is called “fictitious”.

Centrifugal force Centrifugal force

Once you reach the end of the seat, The “centrifugal force” provides a the car door will push on you to sensation that feels very real. keep you in the circle. Be careful when solving circular This ACTUAL force is the Fc motion problems that you are not centripetal force. tricked into including this fictitious Fc force on free-body diagrams.

Circular motion problem Circular motion problem

Federal guidelines suggest that the maximum safe sideways Federal guidelines suggest that the maximum safe sideways acceleration in a turn is 1 m/s2. What is the minimum radius acceleration in a turn is 1 m/s2. What is the minimum radius curve a civil engineer should design on a road where cars travel curve a civil engineer should design on a road where cars travel at 30 m/s (67 mph)? at 30 m/s (67 mph)?

Asked: r

Given: a

Relationships:

Solution:

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Circular motion problem Circular motion problem

Asked: r Asked: r

Given: ac, v Given: ac, v

Relationships: Relationships:

Solution: Solution:

Circular motion problem Circular motion problem

Federal guidelines suggest that the maximum safe sideways Federal guidelines suggest that the maximum safe sideways acceleration in a turn is 1 m/s2. How much force is needed to acceleration in a turn is 1 m/s2. How much force is needed to create this acceleration for a 1200 kg car? create this acceleration for a 1200 kg car?

What is providing this force? static friction

What might happen if the road is icy?

Circular motion problem Circular motion problem

Federal guidelines suggest that the maximum safe sideways The free-body diagram for a car safely acceleration in a turn is 1 m/s2. How much force is needed to rounding a curve shows the friction, create this acceleration for a 1200 kg car? which provides the centripetal force.

SIDE VIEW What is providing this force? static friction Ff What might happen if the road is icy? F F If there is less than 1200 N of friction available, N N VIEW FROM the car will slide off the road, tangent to the circle. mg ABOVE

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Circular motion problem What does “high-g” mean?

The friction must point toward This airplane’s high velocity as it the center of the circle. moves through a tight turn results in a centripetal acceleration of 100 m/s2.

This is about 10 times the acceleration of gravity (g = 9.8 m/s2).

center of circle FN FN mg

What does “high-g” mean? Assessment

This airplane’s high velocity as it 1. A race car is moving with a speed of 200 km/h on a circular section moves through a tight turn results in of a race track that has a radius of 400 m. The race car and driver a centripetal acceleration of 100 m/s2. have a mass of 1400 kg.

This is about 10 times the a) What is the magnitude of the centripetal acceleration felt acceleration of gravity (g = 9.8 m/s2). by the driver?

The centripetal force needed on a Asked: a Solution: 70 kg pilot would be 7000 N. Given: v At this acceleration, the heart cannot pump blood to the brain and the pilot Relationship: would lose consciousness.

Assessment Assessment

1. A race car is moving with a speed of 200 km/h on a circular section 1. A race car is moving with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and driver of a race track that has a radius of 400 m. The race car and driver have a mass of 1400 kg. have a mass of 1400 kg.

a) What is the magnitude of the centripetal acceleration felt a) What is the magnitude of the centripetal acceleration felt by the driver? by the driver?

Asked: ac Solution: Asked: ac Solution:

Given: v, r, m Given: v, r, m

Relationship: Relationship:

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Assessment Assessment

b) What is the centripetal force acting on the mass? b) What is the centripetal force acting on the mass?

Asked: Fc Solution: Asked: Fc Solution:

Given: m Given: m, ac

Relationship: Fc Relationship: Fc = mac

Assessment Assessment

2. In the case of a car rounding a turn, what real physical force 2. In the case of a car rounding a turn, what real physical force is acting radially to keep the car moving in a circular path? is acting radially to keep the car moving in a circular path?

Static friction between the tires and the road provides the centripetal force.