Uniform Circular Motion

Pg. 114 - 130

Uniform Circular Motion

Have you ever ridden on the ride shown below? As it

spins you feel as though you are being pressed tightly

against the wall. And then the floor drops away and the

ride begins to tilt. But you remain “glued” to the wall.

What is unique about moving in a circle that allows you

to apparently defy gravity? What causes people on the

ride to “stick” to the wall?

Uniform Circular Motion

Amusement park rides are only one of a very large

number of examples of circular motion. When an object

is moving in a circle and its speed is constant, it is said to

be moving with uniform circular motion

Uniform Circular Motion

Take note!

Since objects experiencing uniform circular motion are moving

in a circular path, not only is their direction changing but so it

their velocity. As a result, they are accelerating.

Centripetal Acceleration

For example, consider an object as it moves from point P

to point Q as shown. If its velocity changes from Vi to Vf

then:

∆V = Vf – Vi

Using triangle congruencies and the equations

V = ∆d/∆t and a = ∆v/t then we can show:

ac = v2/r

Centripetal Acceleration

Take note!

Since Vi and Vf are perpendicular

to the radii of the circle, the

acceleration vector points directly

toward the centre of the circle.

Acceleration that is directed

toward the centre of a circular

path is called centripetal

acceleration (ac)

(note) Uniform Circular Motion

Occurs when an object moves in a circle and its speed is constant

Since direction changes the object experiences centripetal acceleration

Note:

Centripetal acceleration is always directed toward the centre of the circle

Centripetal Acceleration

Practice:

1. A child rides a carousel with a radius of 5.1 m that rotates

with a constant speed of 2.2 m/s. Calculate the magnitude of

the centripetal acceleration of the child.

Centripetal Acceleration

Sometimes you may not know the speed of an object

moving with uniform circular motion.

However, you may be able to measure the time it takes

for the object to move once around the circle, or the

period (T)

If the object is moving too quickly, you would measure

the number of revolutions per unit time, or the

frequency (f). Recall:

f = 1/T

In each case, the equation for centripetal acceleration

would become:

(note) Centripetal Acceleration (more)

Practice

2. A salad spinner with a radius of 9.7 cm rotates

clockwise with a frequency of 12 Hz. At a given instant, a

piece of lettuce is moving in the westward direction.

Determine the magnitude and direction of the centripetal

acceleration of the lettuce in the spinner at the moment

shown (because, doesn’t everybody wonder how fast their lettuce is

accelerating when making a salad? )

Practice

3. The centripetal acceleration at the end of a fan blade is

1750 m/s2. The distance between the tip of the fan blade

and the centre is 12.0 cm. Calculate the frequency and

the period of rotation of the fan.

Centripetal Force

According to Newton’s laws of motion, an object will

accelerate only if a force is exerted on it.

Since an object moving with uniform circular motion is

always accelerating, there must always be a force exerted

on it in the same direction as the acceleration, as shown:

Centripetal Force

Since the force causing a centripetal acceleration is always

pointing toward the centre of the circular path, it is called

a centripetal force (Fc)

Without such a force, objects would not be able to move

in a circular path

Centripetal Force

Using Newton’s second law and ac= v2/r the formula for

Fc is:

**think of Fc as Fnet when dealing with circular motion

(note) Centripetal Force

Net force that causes centripetal acceleration (Fc = Fnet)

Centripetal Force

Take note:

A centripetal force can be supplied by any type of force

For example, gravity provides the centripetal force that keep

the Moon on a roughly circular path around Earth, friction

provides a centripetal force that causes a car to move in a

circular path on a flat road, and the tension in a string tied to a

ball will cause the ball to move in a circular path when you

twirl it around.

Practice

4. Suppose an astronaut in deep space twirls a yo-yo on a string.

A) what type of force causes the yo-yo to travel in a circle?

(tension)

B) What will happen if the string breaks?

(the yo-yp will move along a straight line, obeying Newton’s first law – objects in motion tend to stay in motion)

5. A car with a mass of 2200 kg is rounding a curve on a level road. If the radius of the curvature of the road is 52 m and the coefficient of friction between the tires and the road is 0.70, what is the maximum speed at which the car can make the curve without skidding off the road?

Practice

6. You are playing with a yo-yo with a mass of 225 g. The full length of the string is 1.2m.

A) calculate the minimum speed at which you can swing the yo-yo while keeping it on a circular path (hint: at the top of the swing Ft= 0)

B) at the speed just determined, what is the tension in the string at the bottom of the swing.

7. A roller coaster car is at the lowest point on its circular track. The radius of curvature is 22 m. The apparent weight of one of the passengers is 3.0 times her true weight (i.e. FN = 3Fg). Determine the speed of the roller coaster

Centripetal Force & Banked Curves

Cars and trucks can use friction as a centripetal force.

However, the small amount of friction changes with road

conditions and can become very small when the roads

are icy

As well, friction causes wear and tear on tires and causes

them to wear out faster

For these reasons, the engineers who design highways

where speeds are high and large centripetal forces are

required incorporate another source of centripetal force

– banked curves

Practice

8. What angle of banking would allow a vehicle to move

around a curve with a radius of curvature “r” at a speed

“v”, without needing any friction to supply part of the

centripetal force? (In this case you must resolve FN so

that one of the components is directed inward.

Centripetal Force & Banked Curves

Take Note:

When an airplane is flying straight and horizontally, the wings

create a life force (L) that keeps the airplane in the air

However, when an airplane need to change directions it must

tilt or bank in order to generate a centripetal force

The centripetal force created is a component of the lift force,

as shown:

Artificial Gravity

On Earth, the gravity we experience is mainly due to

Earth itself because of its large mass and the fact that we

are on it

However, there is no device that can make or change

gravity

So how can we simulate gravity? The answer is simple –

uniform circular motion

Incorporating the principles of uniform circular motion in

technology has led to advance in

many fields, including medicine,

industry, and the space program

Artificial Gravity

For example, making a spacecraft rotate constantly can

simulate gravity.

And, if the spacecraft rotates at the appropriate

frequency, the simulated gravity can equal Earth’s gravity

As a result, many of the problems faced by astronauts

working and living in space, such as bone loss and muscle

deterioration, could be eliminated (or at least minimized)

Textobook:

Pg. 119, #6,8

Pg. 124, #3,4

Pg. 130, #6,7