Visualizing Centripetal Force and Acceleration Centripetal Force – the force that causes an object to travel in a circular path Centripetal Acceleration – the acceleration caused by the force which causes object to travel in a circular path

Why is a force necessary to turn a car? Newton An object in motion will stay in motion with the same velocity & direction unless acted upon by an outside force.

Think about a coffee cup on the dashboard of your car. What happens to the cup as you make a left turn? Cup

Path of cup Path of car

To the driver of the car, the cup appears to move to the right during the left turn, if there were no friction between the cup and the dashboard. The difference in paths is the movement of the cup relative to the car

Now analyze cars traveling on a level surface in right and left turns.

First the Right Turn. If the car shown is to make right turn what direction must the centripetal force act?

If the car shown is to make a right turn what direction must the centripetal force act? The force must also act to the right.

What creates this force?

THE ROAD

Consider what the same situation would look like from above the car.

Centripetal Force Velocity Path of the car

Centripetal Force Velocity What is the angle between the velocity and the Centripetal Force?

Centripetal Force Velocity What is the angle between the velocity and the Centripetal Force? 90 degrees

As car turns, the velocity changes direction. The Force turning the car stays perpendicular to the velocity, always pointing towards the center of the circle.

If the car shown is to make left turn what direction must the centripetal force act?

If the car shown is to make a left turn what direction must the centripetal force act? The force must also act to the left.

Consider what the same situation would look like from above the car.

What do the vectors for velocity and centripetal force look like for the left turn?

Centripetal Force Velocity Left Turn

As car turns, the velocity changes direction. The Force turning the car stays perpendicular to the velocity, always pointing towards the center of the circle.

Turning RightTurning Left

Consider identical cars (equal mass) traveling at the same velocity. Which car will experience the greatest centripetal force?

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string?

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string Radius of circle - as radius the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string Radius of circle - as radius the Tension in the string

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string Radius of circle - as radius the Tension in the string Direct relationship Inverse relationship

The tension in the string provides the Centripetal force necessary to keep the airplane flying in the circle as shown. What physical variables of effect the tension on the string? Velocity - as velocity the Tension in the string Mass of plane - as mass the Tension in the string Radius of circle - as radius the Tension in the string Direct relationship Inverse relationship Centripetal Force Equation

Analyze the Units

You should be able to solve this equation for any one of the variables

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables:

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram:

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation:

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation:

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation: Solve for Desired Variable

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation: Solve for Desired Variable

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation: Solve for Desired Variable Plug in with Units

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation: Solve for Desired Variable Plug in with Units

Example Question: A person holds a 5 kg airplane in a 4 meter radius circle with a tension in the rope of 25 N. What is the velocity of the airplane? Create Diagram: List Variables: m=5 kg, F c = 25 N r = 4 m State Law or Equation: Solve for Desired Variable Plug in with Units

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. At first it appears that you don’t have enough information but by creating a ratio you can eliminate many variables.

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1.

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1.

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1. Simplify by cancelling

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1. Simplify by cancelling

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1.

Example Question where a Ratio is used to find answer: A mass m is traveling in a circular path with radius r at a velocity v. By what factor will the centripetal force change if the mass increases by a factor of 2? (The radius and velocity stay the same. Create a ratio of the forces of the two situations, F 2 / F 1. Rearrange for an alternate form of this relation.

Here is another example illustrating how the area of a circle changes as radius increases: CAREFUL, ITS NOT ALWAYS AS SIMPLE AS THE LAST ONE The radius of a circle increases by a factor X. By what factor does the area of the circle change?

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1.

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1.

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Remember that r 2 = Xr 1

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Remember that r 2 = Xr 1

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Simplify

Here is another example illustrating how the area of a circle changes as radius increases: The radius of a circle increases by a factor X. By what factor does the area of the circle change? Create a ratio of the areas of the two situations, A 2 / A 1. Rearrange

5.4 Banked Curves On an unbanked curve, the static frictional force provides the centripetal force.

Now consider objects turning in a vertical plane:

Car drawing Reference: