Causes of Circular MotionChapter 7 Section 3

Force That Maintains Circular Motion

When an object is moving in a circular path, it has a centripetal acceleration pointing towards the axis.

The inertia of the object tends to make the object’s motion in a straight-line path, but what keeps it in the circular path? FORCE!

Force and Circular Motion

A force makes the object follow the circular path and is directed towards the axis.

This force can be found by applying Newton’s Second Law along the radial direction.

Centripetal Force Equation

Fc = Centripetal Force (N) m = Mass (kg) r = Radius (m) Vt = Tangential Speed (m/s) ω = Angular Speed (rad/s)

In order to calculate the correct force, the correct units must be used in the equation!

𝐹 𝑐=𝑚𝑣𝑡

2

𝑟

Centripetal Force Explained

A centripetal force can be generated by other forces, such as gravity or an object attached to a string that is subjected to circular motion. Gravity, friction, tension, normal force, etc

Any force can produced a centripetal force if is follows certain rules.

Force Direction

Forces pointing inward toward the axis are considered positive. (+)

Forces pointing outward away from the axis are considered negative. (-)

+ --

-

-

g-Force Equations

What is a g-Force A g-force can be related through

accelerations or forces. Acceleration due to gravity (9.8m/s²)

1g = 9.8m/s² Greater centripetal accelerations, yield

greater g-forces. Weight of a person standing still

1g = weight of person at rest Acceleration produces forces and a g-force

Elevator going up makes you feel heavier Elevator going down makes you feel lighter

g-Force Chart

g-Force Force Felt

g-Force = 1g Feel normal

g-Force > 1g Feel heavier then normal

0g < g-Force < 1g Feel lighter then normal

g-Force = 0g Feel weightlessness

g-Force < 0g Feel like floating away

A Force is Need to Keep an Object in Circular Motion

A force directed towards the center is necessary to keep an object in circular motion.

If the force vanishes, the object will not continue in its circular path and will travel in a straight line path from the time the force disappears.

Describing the Motion of a Rotating System

Imagine a car and taking a turn real hard to the left.

A person sitting in the passenger's seat tends to slide to the right and hits the door.

What causes the person to slide up against the door?

Centrifugal Force (Imaginary)

A common misconception is that there is a mysterious force that pushes the person up against the door. This is called a Centrifugal Force

Centrifugal – means “Inter-fleeing” or “Outer-seeking”

Centrifugal Forces does not exist!!!

Inertia Inertia is why the person slides towards

the door. Inertia – The tendency for an object to

resist change. An object in motion, tends to stay in a constant state of motion in a straight line path.

Follows from Newton’s First Law of Motion

A force is needed to change the state of motion of an object and that force is a Centripetal Force

Inertia and Forces

Inertia is often misinterpreted as a force. Inertia is what makes an object tend to

move in a straight line path, but the Centripetal force pulls the object inward towards the axis and keeps it in a circular path.

Example Problem #1

An astronaut who weights 735N on Earth is at the rim of a cylindrical space station with a 73m radius. The space station is rotating at an angular speed of 3.5rpm. Evaluate the force that maintains the circular motion of the astronaut.

Example Problem Answer #1

Fc = mrω²

= (735N/9.8m/s²)(73m)(3.5rev/min)² Convert rev/min to rad/s

= (75kg)(73m)(.37rad/s)²

= 749.53N

Fc = 749.53N

Example Problem #2

A person sitting in a car who’s weight is 650N is driving in a car at 25m/s in a circular path of radius 50m. Find the centripetal force keeping the person in the circular path. What is the g-force on the person?

Example Problem Answer #2

Fc = 829.08N g-force = 1.28g

Example Problem #3

A car rounds a curve on a level road that has a radius of 200m. The coefficient of friction between the road and tires of the car is 0.80. What is the max speed at which the car can travel around the curve without losing control of the car? What is the max speed if the coefficient was dropped to 0.20 (snow covered road)?

Example Problem Answer #3

Coefficient of 0.80 = 39.60 m/s

Coefficient of 0.20 = 19.80 m/s

Example Problem #4

A looping rollercoaster ride at an amusement park has a radius of 10m. What is the minimum velocity the coaster must be moving in order for the passengers to stay in their seats as it reaches the top of the arch?

Example Problem Answer #4

= 9.90m/s

Example Problem #5

An engineer is designing an exit ramp to an interstate highway. The radius of the curve is 75m and is banked at 15 degrees. What should the max speed limit be on the exit ramp assuming the road is covered in ice and has no friction?

Example Problem #5 Setup

Example Problem Answer #5

= 14.03m/s

Gravitational Forces

Gravitational Force – The mutual force attraction between particles of matter. All objects are attracted to one another

through a gravitational force. The planets travel in a near circular orbit

around the sun. The force that keeps the planets from coasting off in a straight line path is a gravitational force.

Gravitational Field Force

Gravitational forces is a field force that exist between two masses regardless of the medium that separates them.

It does not matter how small or how large the masses are, there is a gravitational force between the two forces.

Funny Video

Newton’s Third Law

Gravitational Forces follow Newton’s Third Law of Motion.

Ex: Gravity Pulls down on the Apple and the

Apple pulls up on the Earth. Another Funny Video

Newton’s Universal Law of Gravitation

= Gravitational Force = mass #1 = mass #2 = Distance between the center of masses = Gravitational Constant

𝐹 𝑔=𝐺𝑚1𝑚2

𝑟2

Gravitational Constant

“G” is a universal constant called Constant of Universal Gravitation

It is a constant that is used to calculate gravitational forces between any two masses.

𝐺=6.673 𝑥10−11𝑁𝑚2

𝑘𝑔2

Inverse-Square Law

The force between two masses varies as the inverse square of the separation.

As the distance between the two masses increase, the force between the masses decrease.

Center of Mass

The Gravitational Force is localized to the center of a spherical mass.

If not spherical, the center of mass is used.

Satellite Motion

Satellite motion can be described through projectile motion.

The equation for satellite motion around a planet can be solved through the use of Newton’s Laws, Circular motion, and Newton’s Universal Graviton Law.

𝑣=√𝐺𝑀 𝐸

𝑅+𝑟

Example Problem #6

What is the gravitational force between the Earth and the Moon? Radius of Earth = 6371 km Mass of Earth = 5.98x10^24 kg Radius of Moon = 1737 km Mass of Moon = 7.36x10^22 kg Distance between Earth and Moon =

3.76x10^8 m

Example Problem Answer #6

Fg = 1.99x10^20 N

Example Problem #7

A satellite is sent into space at a distance of 36,000km above the earth’s surface. At what speed should the satellite be moving in order to stay in orbit?

Example Problem Answer #7

= 3,068.89m/s