 # Circular and Centripetal Motion

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Circular and Centripetal Motion

Moving objects can be described by using kinematic equations.
The motion of moving objects can be explained by Newton’s Laws These principles can be applied to circular motion as well.

Remember Newton’s 1st Law?
An object at rest tends to stay at rest while an object in motion tends to stay in motion at constant velocity… …unless acted on by an outside force.

Circular motion in not natural. A force is required to change direction. A change in direction means a change in velocity… which means there is an acceleration. The acceleration comes from the applied force… uniform circular motion is caused by an applied force.

Uniform Circular Motion
Motion of an object in a circular pattern with constant velocity It is only one of the forms of circular motion. An object moving in a circle will cover the same linear distance each second. Example: a car moving in a circle at a constant speed of 5 m/s will traverse 5 meters around the perimeter per second. Circumference: distance of one complete cycle around the perimeter

Average speed = distance/time
Average speed = circumference/time Circumference = 2 x π x R / T Period: one cycle around the circle Average speed = 2πR/ T R = radius of the circle; T = period

The average speed and the radius are directly proportional.
For objects moving around a circle with different radii, in the same period, the object travelling the greatest radius has the greatest speed. The average speed and the radius are directly proportional. If the radius doubles, the speed will double.

The Velocity Vector If all objects moving in a circular motion have a constant speed, will they have a constant velocity? Review: vector vs. scalar The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the vector is in the same direction as the object moves. Since the object is moving in a circle, its direction is constantly changing.

vector is the same but the direction is constantly
The magnitude of the vector is the same but the direction is constantly changing with position. A better way to describe the direction of the velocity vector is tangential. At any given instant its direction is in the same direction as a tangent drawn to the circle. v v v v

If an object moving in uniform circular motion, is there acceleration? An accelerating object is one that is changing its velocity. Velocity is a vector with magnitude (speed) and direction so a change in either results in a change in velocity. In this case, direction is changing so there is a change in velocity and therefore acceleration.

Formula review….. Acceleration = change in velocity /change in time Or
a = ∆v / t ∆= vfinal – vinitial Vi vf

In the case of circular motion, the acceleration is directed toward the center of the circle. According to Newton’s second law, F = ma, an object that is accelerating but be experiencing a net force. The direction of the net force is in the same direction as the acceleration. Therefore for an object moving in a circle there must be an inward force acting on the object to cause inward acceleration. This is known as the centripetal force requirement. (means toward the center or center seeking) Note: Centripetal is not the same as centrifugal.

Centripetal vs. Centrifugal
Centripetal force: the force that keeps the object moving in a circle Centrifugal force: the force you “feel is being exerted on you” … doesn’t actually exist… How you feel in a turning car How you feel at the top of a roller coaster

Review Newton’s First…
The Law of Inertia: An object at rest remains at rest and an object in motion will remain in motion at the same speed and direction unless acted upon by an unbalanced force. Moving objects tend to move in straight lines. Therefore for an object to move in a circle, there must be an unbalanced force present.

Consider riding in a car or on a roller coaster
Consider riding in a car or on a roller coaster. The car begins to turn to the right. What happens to the passenger (blue) in the car as the driver (red) goes around a right curve or makes a right turn? What happens to the driver?

Examples: A car turning a corner
Force provided by friction between tires and road

As a car makes a turn, the force of friction acting upon the turned wheels of the car provides centripetal force required for circular motion. As a bucket of water is tied to a string and spun in a circle, the tension force acting upon the bucket provides the centripetal force required for circular motion. As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

Examples: A ball twirled at the end of a string Tetherball
Force provided by tension in string Tetherball

Examples: Water that stays in a swinging bucket
Force caused by the bottom of the bucket, or the normal force Water in glass Demo

Examples: Earth orbiting around the sun Solar System
Force provided by gravitational attraction between two bodies Solar System

What is the difference in linear and centripetal forces?
Forces that cause things to speed up we will call linear forces. Forces that cause things to change direction we will call centripetal forces.

Can there be both? Sure… a car can speed up around a corner, or slow down as it turns.

Let’s draw a representation of the vectors involved
m

Uniform Circular Motion
For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration 24

Uniform Circular Motion
For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration 25

Period: the time it takes for an object to complete one cycle, in this case a rotation, T
Frequency: the number of times an object completes a cycle in a given amount of time i.e. seconds Hertz: the unit to measure frequency instead of saying s^-1 … 1/s = Hz

After closing a deal with a client, Kent leans back in his swivel chair and spins around with a frequency of 0.5 Hz. What is Kent’s period of spin?

Rotational speed: # of degrees or radians an object in circular motion goes through per second
Linear speed: speed of an object in one direction Tangential speed: the speed of an object at any point on the edge of the circle (linear speed for circular motion)

What equation can we use to measure the speed of an object as it travels in circular motion?
-How far does something travel to get all they way around a circle? -What do we call the time it takes to go that far?

Speed = Δ distance / Δ time
The time for an object to complete one revolution is its period T… The distance traveled in one revolution is r The speed of an object would then be… 30

Curtis’ favorite record has a scratch 12 cm from the center that makes the record skip 45 times each minute. What is the linear speed of the scratch as it turns?

Let’s look at what’s happening mathematically…

Uniform Circular Motion
Newton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body. *The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle. 33

Equations

Lets try one together… A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the stopper.

What if… …the mass is doubled? …the radius is doubled?
…the period is halved?

Missy’s favorite ride at the Topsfield Fair is the rotor, which has a radius of 4.0 m. The ride takes 2.0 s to make one full revolution. A) What is Missy’s linear speed on the rotor? B) What is Missy’s centripetal acceleration on the rotor?

As their booster rockets separate, space shuttle astronauts typically feel accelerations up to 3g, where g = 9.80 m/s2. In their training, astronauts ride in a device where they experience such an acceleration as a centripetal acceleration. Specifically, the astronaut is fastened securely at the end of a mechanical arm, which then turns at a constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of 3.00g while in circular motion with radius 9.45 m.

An object of mass kg is attached to the end of a cord whose length is 1.50 m. The object is whirled in a horizontal circle. If the cord can withstand a maximum tension of 50.0N, what is the maximum speed the object can have before the cord breaks?

Lets try a problem or two, or three……
The Texas Star Ferris wheel has a radius of 32 meters. When operating with constant tangential velocity, it completes one rotation in 2 minutes. A 60 kg rider sits on the bench in one of the wheel’s baskets. What is the rider’s centripetal acceleration? How big is the centripetal force required to keep the rider going in the big circle at that speed?