1. Uniform Circular Motion
Physics Notes Ch. 9
Uniform Circular Motion

2. 9.1 Uniform circular motion
Uniform circular motion: Movement in a circle at a constant speed.
Ball on a string - If speed is constant – constant circular motion – if speed changes - variable circular motion.
Circles
Uniform circular motion

3. 9.2 Period
Period: The amount of time it takes for an object to return to the same position.
The period is the circumference of the circle, 2πr, divided by the object’s speed or distance divided by speed.
T=2πr/v
T=period s
R=radius m
V=velocity m/s
Period

4. 9.3 Interactive checkpoint: a spinning CD
Period and Frequency
The Period (T) of a body travelling in a circle at constant speed is time taken to complete one revolution - measured in seconds
Frequency (f) is the number of revolutions per second – measured in Hz
9.3 Interactive checkpoint: a spinning CD
T = 1 / f f = 1 / T

5. Going round in circles Speed may be constant
But direction is continually changing
Therefore velocity is continually changing
Hence acceleration takes place

6. 9.4 Centripetal Acceleration
Centripetal acceleration: The centrally directed acceleration of an object due to its circular motion.
Change in velocity is towards the center
Therefore the acceleration is towards the center
This is called centripetal acceleration
for circular motion in general, there may be both centripetal acceleration, which changes the object’s direction, and acceleration in the direction of the object’s motion (tangential acceleration), which changes its speed.
ac=V2/r
Centripetal Acceleration

7. 9.5 Interactive problems: racing in circles

8. 9.6 Newton's second law and centripetal forces
Acceleration is caused by Force (F=ma)
Force must be in the same direction as acceleration
Centripetal Force FC acts towards the center of the circle
Centripetal force is provided by some external force – e.g. Friction
Fc =mv2/r
Centripetal force
9.7 Sample problem: banked curves
9.8 Sample problem: centripetal force on a pendulum

9. Force and Acceleration
v = 2π r / T and T = 2π / ω
v = r ω
a = v² / r = centripetal acceleration
a = (r ω)² / r = r ω² is the alternative equation for centripetal acceleration
FC = m r ω² is centripetal force

10. Examples of Centripetal Force
Friction
Tension in string
Gravitational pull
Friction
velocity
Tension
velocity
Weight

11. Centripetal Force cont…
What provides the centripetal force in each case ?

12. Centripetal force cont…

13. Circular Motion Calculations
Centripetal acceleration
Centripetal force

14. 9.9 Accelerating reference frames and fictitious forces
A system for describing the location of objects.
Inertial Frame of Reference – Newton’s First Law holds true.
Noninertial Frame of reference – Body is accelerating so Newton’s First Law doesn’t hold true
Fictitious force: A perception of force caused by the acceleration of a reference frame.
Centrifugal Force – a force that tends to move the particles of a spinning body away from the spin axis (this force only exists for an observer in a non-inertial frame of reference). THIS IS A FICTICIOUS FORCE!
Accelerating reference frame

15. Weightlessness Red Arrows pilots experience up to 9g (90m/s²)
These astronauts are in freefall
Red Arrows pilots experience up to 9g (90m/s²)
This rollercoaster produces accelerations up to 4g (40m/s²)

16. Weightlessness
True lack of weight can only occur at huge distances from any other mass
Apparent weightlessness occurs during freefall where all parts of you body are accelerating at the same rate

17. 9.10 Artificial gravity
Why does the rotation of a spacecraft produce the sensation of gravity? Consider what happens when an airplane takes off from a runway: You feel a force pulling you back into your seat, as if the force of gravity were increasing. The force of gravity has not been significantly altered (in fact, it decreases a bit as you gain elevation). However, while the airplane accelerates upward, you feel a greater normal force pushing up from your seat, and you may interpret this subconsciously as increased gravity.
Artificial gravity is a pseudo, or fictitious, force it would disappear if the spacecraft stopped rotating.
Artificial gravity

18. Motion in Vertical Circle
νmin = velocity at top of circle
νmax= velocity at bottom of circle
Critical Velocity – the minimum velocity necessary to maintain uniform motion in a vertical circle. It is independent of mass

19. 9.11 Loop-the-loop Loop-the-loop
Loop the loop is possible if the track provides part of the Fc at the top of the loop ( ST )
The rest of the Fc is provided by the weight of the rider
Loop-the-loop
9.12 Interactive checkpoint: maximum loop-the-loop radius
9.13 Interactive summary problem: race curves