SPH4U UNIVERSITY PHYSICS - Meant4Teachers.commeant4teachers.com/wp-content/uploads/2013/09/Uniform-Circular... · SPH4U UNIVERSITY PHYSICS ... gravity provides the centripetal force

24/09/2012

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SPH4UUNIVERSITY PHYSICS

DYNAMICS

L Uniform Circular Motion

(P.114-130)

Uniform Circular Motion

Have you ever ridden on the ride shown below? As it spins you feel asthough you are being pressed tightly against the wall. And then the floordrops 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 defygravity? What causes people on the ride to “stick” to the wall?

September 24, 2012 4U1 - Uniform Circular Motion 1


Amusement park rides are only one of a very large number of examples ofcircular motion. When an object is moving in a circle and its speed isconstant, it is said to be moving with uniform circular motion.


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NOTE!

Since objects experiencing uniform circular motion are moving in a circularpath, not only is their direction changing but so too is their velocity. As aresult, they are accelerating.


Centripetal Acceleration

For example, consider an object as it movesfrom point P to point Q as shown. If itsvelocity changes from v1 to v2 then:

)v = v2 – v1

Using triangle congruencies and theequations v = )d/)t and a = )v/t then wecan show:

ac = v 2 / r



NOTE!

Since v1 and v2 are perpendicular to theradii of the circle, the acceleration vectorpoints directly toward the centre of thecircle. Acceleration that is directed towardthe centre of a circular path is calledcentripetal acceleration (ac ).


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UNIFORM CIRCULAR MOTION

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

� since direction changes the object experiences centripetal acceleration

where ac is the centripetal acceleration (m/s2)

v is the velocity (m/s)

r is radius of rotation (m)

NOTE!

Centripetal acceleration is always directed toward the centre of the circle.


r

v a

2

c =


PRACTICE

1. A child rides a carousel with a radius of5.1 m that rotates with a constant speedof 2.2 m/s. Calculate the magnitude ofthe centripetal acceleration of the child.

ac = 0.95 m/s2



Sometimes you may not know the speed ofan object moving with uniform circularmotion. However, you may be able tomeasure the time it takes for the object tomove once around the circle, or the period(T). If the object is moving too quickly, youwould measure the number of revolutions perunit time, or the frequency (f) . Recall:

f = 1/T

In each case, the equation for centripetalacceleration would become:


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CENTRIPETAL ACCELERATION (more ...)

where ac is centripetal acceleration (m/s2)

r is the radius of rotation (m)

T is the period of rotation (s)

f is the frequency of rotation (Hz)


22

2

2

c rf4or T

r4 a ππ=


PRACTICE

2. Explain how the following formulas are derived? (Hint: d = C = 2Br)


22c2

2

c rf4 a & T

r4 a π=π=


PRACTICE

3. A salad spinner with a radius of 9.7 cmrotates clockwise with a frequency of12 Hz. At a given instant, a piece oflettuce is moving in the westwarddirection. Determine the magnitudeand direction of the centripetalacceleration of the lettuce in thespinner at the moment shown.

ac = 550 m/s2[N]


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PRACTICE

4. 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 12.0cm. Calculate the frequency and the period of rotation of the fan.

f = 19.2 Hz

T = 0.0520 s


Centripetal Force

According to Newton’s laws of motion, an objectwill accelerate only if a force is exerted on it.Since an object moving with uniform circularmotion is always accelerating, there mustalways be a force exerted on it in the samedirection as the acceleration, as shown.


Centripetal Force

Since the force causing a centripetalacceleration is always pointing toward thecentre of the circular path, it is called acentripetal force (Fc). Without such a force,objects would not be able to move in a circularpath.


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Centripetal Force

Using Newton’s second law and ac = v 2/r theformula for Fc is:


r

mv F

ma F

ma F ΣF

2

c

cc

net

=

==

NOTE!Think of Fc as Fnet when dealing with circular motion.

Centripetal Force

CENTRIPETAL FORCE

� net force that causes centripetal acceleration (Fc =Fnet)

where Fc is the centripetal force (N) 7 acts toward centre of circle

m is the mass (kg)

NOTE!

Always choose motion towards the centre of the circle as the +ve direction.


T

mr4 mrf4

r

mv F

2

222

2

c

ππ ===

Centripetal Force

NOTE!

A centripetal force can be supplied by any type of force. For example,gravity provides the centripetal force that keeps the Moon on a roughlycircular path around Earth, friction provides a centripetal force that causesa car to move in a circular path on a flat road, and the tension in a stringtied to a ball will cause the ball to move in a circular path when you twirl itaround.


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Centripetal Force

PRACTICE

5. Suppose an astronaut in deep spacetwirls a yo-yo on a string.

(a) What type of force causes the yo-yoto travel in a circle?

(a) tension


Centripetal Force

PRACTICE

5. Suppose an astronaut in deep spacetwirls a yo-yo on a string.

(b) What will happen if the stringbreaks?

(b) The yo-yo will move along astraight line, obeying Newton’s firstlaw.


Centripetal Force

PRACTICE

6. A car with a mass of 2200 kg is rounding a curve on a level road. Ifthe radius of the curvature of the road is 52 m and the coefficient offriction between the tires and the road is 0.70, what is the maximumspeed at which the car can make the curve without skidding off theroad?

Hint: EF Fc = mac

v = 19 m/s (68 km/h)


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Centripetal Force

PRACTICE

7. You are playing with a yo-yo with a mass of225 g. The full length of the string is 1.2 m.

(a) Calculate the minimum speed at which youcan swing the yo-yo while keeping it on acircular path.

(a) Hint: at the top of the swing FT = 0

v = 3.4 m/s


Centripetal Force

PRACTICE

7. You are playing with a yo-yo with a mass of225 g. The full length of the string is 1.2 m.

(b) At the speed just determined, what is thetension in the string at the bottom of theswing.

(b) Hint: EF Fc = FT + Fg

FT = 4.4 N


Centripetal Force

PRACTICE

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

v = 21 m/s


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Centripetal Force & Banked Curves

Cars and trucks can use friction as a centripetal force. However, theamount of friction changes with road conditions and can become very smallwhen the roads are icy. As well, friction causes wear and tear on tires andcauses them to wear out faster. For these reasons, the engineers whodesign highways where speeds are high and large centripetal forces arerequired incorporate another source of centripetal force – banked curves.



PRACTICE

9. What angle of banking would allow avehicle to move around a curve with aradius of curvature “r” at a speed “v”,without needing any friction to supplypart of the centripetal force? (In thiscase you must resolve FN so that one ofthe components is directed inward.)

tan2 = v2/rg



NOTE!

When an airplane is flying straight and horizontally, the wings create a liftforce (L) that keeps the airplane in the air. However, when an airplaneneeds to change directions it must tilt or bank in order to generate acentripetal force. The centripetal force created is a component of the liftforce, as shown.


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Artificial Gravity

On Earth, the gravity we experience is mainly due to Earth itself because ofits large mass and the fact that we are on it. However, there is no devicethat can make or change gravity. So how can we simulate gravity? Theanswer is simple – uniform circular motion. Incorporating the principlesof 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 simulatedgravity can equal Earth’s gravity. As a result, many of the problems facedby astronauts working and living in space, such as bone loss and muscledeterioration, could be eliminated (or at least minimized).


U Check Your Learning

TEXTBOOK

P.119 Q.6,8 (Review)



WIKI (DYNAMICS)

• 4U1 - QUIZ#4 (Projectile & Centripetal Motion)

September 24, 2012 294U1 - Uniform Circular Motion

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SPH4U UNIVERSITY PHYSICS - Meant4Teachers.commeant4teachers.com/wp-content/uploads/2013/09/Uniform-Circular... · SPH4U UNIVERSITY PHYSICS ... gravity provides the centripetal force