Uniform Circular Motion

Uniform Circular Motion

With Newton's Universal Law of Gravitation too!!!

• Something going in a circle that has...

• Constant radius

• Constant speed

​

​What is UCM?

• The time to go around the circle once is called the period.

• Symbol is T

• Unit is seconds

• It's the seconds per cycle/revolution etc.

​

​Some basics

• The inverse of this idea is frequency.

• Symbol is f

• Unit is 1/sec or a Hertz (Hz)

• It's the cycles/revolutions per second.

​

​Some basics

• Remember...

• 1 revolution = 360 degrees = 2πrads

​

​Some basics

• Since v = x/t

• If we go around once, x is the circumference and..

• t is the period

• giving....

• v = C/T or

• v = Dπ/T or 2πr/T

• or 2πrf

​

​Velocity

• Velocity always points tangent to the circle.

• Called tangential velocity

• Acceleration always points towards the center

• Called centripetal acceleration

​

Direction of vectors

• Centripetal acceleration has a special equation.

• a = v²/r

• REMEMBER! Anything in UCM is accelerating even if its speed is constant because

• It is changing direction and

• Therefore changing velocity

​

Acceleration

Multiple Choice

A racing car is moving around the circular track of radius 300 meters shown above. At the instant

when the car's velocity is directed due east, its acceleration is directed due south and has a magnitude of 3 meters per second squared. When viewed from above, the car is moving

• clockwise at 30 m/s

• clockwise at 10 m/ s

• counterclockwise at 30 m/ s

• counterclockwise at 10 m/s

• with constant velocity

Multiple Choice

A child has a toy tied to the end of a string and whirls the toy at constant speed in a horizontal

circular path of radius R. The toy completes each revolution of its motion in a time period T. What is

the magnitude of the acceleration of the toy?

• Zero

• $$\frac{4\pi^2R}{T^2}$$

• $$\frac{\pi R}{T^2}$$

• g

• $$2\pi g$$

Multiple Choice

A figure of a dancer on a music box moves counterclockwise at constant speed around the path shown above. The path is such that the lengths of its segments, PQ, QR, RS, and SP, are equal. Arcs QR and SP are semicircles. Which of the following best represents the magnitude of the dancer's acceleration as a function of time t during one trip around the path, beginning at point P ?

Multiple Choice

Inside a washing machine, the radius of the cylinder where the clothes sit is 0.50 m. In one of its

settings the machine spins the cylinder at 2.0 revolutions per second. What is the acceleration of an

item of clothing?

• 0.080 m/s/s

• 1.6 m/s/s

• 8.0 m/s/s

• 79 m/s/s

• To accelerate something towards the center of the circle...

• There must be a net force towards the center of the circle.

• Called centripetal force

• Since ΣF = ma_C

• F_C = mv²/r

​

​Force

Multiple Choice

An automobile moves at constant speed down one hill and up another hill along the smoothly curved

surface shown above. Which of the following diagrams best represents the directions of the velocity

and the acceleration of the automobile at the instant that it is at the lowest position. as shown?

Multiple Choice

An object moves at a constant speed in a circular path. Which of the following statements is/are true?

I. The velocity is constant

II. The centripetal acceleration is constant.

III. The net force on the object is zero.

• I only

• II only

• III only

• I and II

  only

• II and III

  only

Multiple Select

A child whirls a ball at the end of a rope, in a uniform circular motion. Which of the following statements is true? Select two answers.

• The speed of the ball is constant

• The velocity is of the ball is constant

• The magnitude of the ball's acceleration is constant

• The net force on the ball is directed radially outwards

Multiple Choice

A ball attached to a string is whirled around in a horizontal circle having a radius r. If the radius of the circle is changed to 4r and the same centripetal force is applied by the string, the new speed of the ball is which of the following?

• One-quarter the original speed

• One-half the original speed

• The same as the original speed

• Twice the original speed

• Four times the original speed

Multiple Choice

A car with speed v and an identical car with speed 2v both travel the same circular section of an

unbanked road. If the frictional force required to keep the faster car on the road without skidding is

F, then the frictional force required to keep the slower car on the road without skidding is

• 4F

• 2F

• F

• 1/2 F

• 1/4 F

• We will use the FBD to set up the 2nd Law with these rules.

• Count as + any force that points towards the center.

• Count as negative any force that points away from the center.

• Ignore any forces that point tangent to the circle

​

Rules for Free Body Diagrams in UCM

• Never never never label Fc on a FBD!!!!!!

• Also the bigger force is ALWAYS the one pointing towards the center.

• That's why it's going in a circle in the first place!

​

Rules for Free Body Diagrams in UCM

• At the top both Fg and T point towards the center so...

• Fg + T = mv²/r

• At the bottom, T points towards the center and Fg away so..

• T - Fg = mv²/r

​

​Ex 1: Bucket swung in a vertical circle.

Multiple Choice

An object weighing 4 newtons swings on the end of a string as a simple pendulum. At the bottom of

the swing, the tension in the string is 6 newtons. What is the magnitude of the centripetal

acceleration of the object at the bottom of the swing?

• 0

• 0.5 g

• g

• 1.5 g

• 2.5 g

Multiple Choice

A 4.0 kg mass is attached to one end of a rope 2 m long. If the mass is swung in a vertical circle from the free end of the rope, what is the tension in the rope when the mass is at its highest point if it is moving with a speed of 5 m/s?

• 5 N

• 10 N

• 40 N

• 90 N

​Ex 1: Bucket swung in a vertical circle.

Multiple Select

A ball attached to a light string swings in a counterclockwise vertical circle, as shown above. Which

of the following arrows represent one of the forces exerted on the ball at the moment it passes

through point P ? Select two answers.

​Ex 2: Roller coaster

Multiple Choice

A 100 kg cart goes around the inside of a vertical loop of a roller coaster. The radius of the loop is 3

m and the cart moves at a speed of 6 m/s at the top. The force exerted by the track on the cart at the top of the loop is

• 200 N

• 800 N

• 1000 N

• 1200 N

• 2200

Multiple Choice

A cart of mass is moving with speed on a smooth track when it encounters a vertical loop of radius as shown above. The cart moves along the inside of the entire loop without leaving the track. All frictional forces are negligible.

Which of the following must be true for the cart to remain on the track when it is at point ?

• The net force exerted on the cart must be less than the force that the track exerts on the cart.

• The net force exerted on the cart must be equal to or greater than the weight of the cart.

• The track must exert an upward force on the cart to prevent it from falling.

• The track must exert a force on the cart that is equal to the weight of the cart.

​Ex 3: Car "hill hoppin'

Multiple Choice

The figures show a cart moving over the top of a hill (Case 1), moving at the bottom of a dip (Case 2), and moving at the top of a vertical loop (Case 3). In each case, the normal force acting on the car is N and the weight of the car is W. In which case is it

A) always true that N > W,

B) and in which case is it always true that W > N ?

• A) Case 1

  B) Case 3

• A) Case 2

  B) Case 1

• A) Case 2

  B) Case 3

• A) Case 3

  B) Case 1

Ex 4: Car turning on road

Multiple Choice

A horizontal disk rotates at a constant angular speed ω. As viewed from above, a coin of mass m on the disk moves counterclockwise in a circle of radius R as shown. Which of the following vectors best represents the direction of the frictional force

exerted on the coin by the disk when the coin is at the position shown?

Multiple Choice

A horizontal disk rotates at a constant angular speed ω. As viewed from above, a coin of mass m on the disk moves counterclockwise in a circle of radius R as shown. Which of the following expressions correctly represents the minimum coefficient of

static friction required to prevent the coin from sliding on the disk?

• Rω²/g

• ω²/Rg

• Rω²/mg

• mRω²

Ex 5: Person on the Gravitron

Tx points towards the center.
Ty and Fg are tangent so ignore.
Tx = Tsinθ = mv²/r
But remember Ty = Fg
So Tcosθ = mg and T = mg/cosθ

​Subbing gives mgtanθ = mv²/r


Be careful, they could use the other angle

​Ex 6: Bucket swung in horizontal circle, dipping slightly

• They could just pretend the object does not dip.

• In that case all of T points towards center

• So just plain ol' T is the centripetal force.

​Ex 6b: Bucket swung in horizontal circle, not dipping

Multiple Choice

A child has a toy tied to the end of a string and whirls the toy at constant speed in a horizontal circular path of radius R. The toy completes each revolution of its motion in a time period T. What is the magnitude of the acceleration of the toy?

• Zero

• 4π²R / T²

• πR / T²

• g

• 2πg

• We did a lab with an idealized set up like the picture to the right.

• The tension is still the Fc but..

• In this case the tension is also equal to the weight of the hanging mass, M.

• So we can say Mg = mv²/r

​Ex 6b: Cont.

Multiple Choice

A student has one end of a light string attached to an object of mass m, and the other end of the string is passed through a tube and attached to an object of mass M. The student swings mass m in a horizontal circle while mass M remains at a constant height. The period is T. Which of the following represents an equation to experimentally determine the gravitational field strength of Earth?

• g = 4π²m₀L / MT²

• g = 4π²ML / m₀T²

• g = 4π²m₀ / MLT²

• g = m₀L / 4π²MT²