4  Circular Motion  IB Physics Content Guide 
Big Ideas
• Objects moving in a circle are experiencing acceleration since the direction of the velocity is constantly changing
• Centripetal acceleration and centripetal force are always directed toward the center of the circle
• The net force for a body in circular motion is equal to the centripetal force
• It is useful to draw a free body diagram to determine what forces are present at a given position
Content Objectives
4.1 – Defining Circular Motion  
I can convert between angular displacement in revolutions and radians  
I can describe and calculate the properties of period and frequency  
I can calculate angular velocity  
I can describe and calculate tangential velocity based on the angular velocity and radius  
I can determine the direction and magnitude of centripetal acceleration and centripetal force  
4.2 – Vertical Circular Motion  
I can draw correctly proportioned free body diagrams for horizontal and vertical circular motion  
I can compare the forces on an object at different positions in vertical circular motion  
I can identify the combination of forces that make up the net force that results in circular motion.  
I can determine the magnitude and direction of the forces needed to move in a vertical circle  
4.3 – Circular Motion, Friction, and Angles  
I can draw a free body diagram when circular motion is produced by a reaction or friction force  
I can solve problems that involve friction to create circular motion  
I can solve circular motion problems that incorporate components of an angled force  
4  Circular Motion  Shelving Guide 
 Symbol  Unit 
 Draw in vectors for v, a_{c}, and F_{c} à  
Distance  d  m 
 
Angular Distance  θ  rad 
 
Angular Velocity  ω  rad s^{1} 
 Data Booklet Equations:  
Linear Velocity  v  m s^{1} 
 
Centripetal Acceleration  a  m s^{2} 
 
Centripetal Force  F_{c}  N 

Defining Circular Motion
Period  T  s  Angular Velocity  ω  rad s^{1}  
Time per revolution 
Vertical Circular Motion
Top:
 Bottom:  
F_{net} = F_{c} = F_{T} + F_{g}  F_{net} = F_{c} = F_{T} – F_{g}  


 
Top:  Bottom:  
F_{net} = F_{c} = F_{g} – R  F_{net} = F_{c} = R – F_{g}  
Circular Motion with Friction and Angles
Relationships between variables:

Relationships between variables:

Relationships between variables:

6.1 Circular Motion
Definition: Moving in a perfect circle, while velocity has a constant magnitude but changing direction.
Quantities
Angular displacement (θ): Angle through which the object moves.
Measured in degrees (º) or radians. 2π radians = 360º.
Angular speed (ω): Δθ/Δt.
Period (T): time taken to complete one revolution.
Link between linear and circular quantities: s = θr and v = ωr, where r is the radius.
Centripetal acceleration (ac)
Object moving in a circle:
 Equation: ac = Δv/Δt = vΔ/Δt = vω = v^2/r = 4rπ^2/T^2.
Reason: Since the velocity is changing the direction when an object moves in a circle, there must be an acceleration.
Direction: Always directed towards the center of the circle. It generates the centripetal force, which is also always directed towards the center.
Centripetal force (Fc)
Equation: Fc = mac = (mv^2)/r = mrω^2. No work, as F is perpendicular to v!
Cases:
Satellites in orbit: Centripetal force = Gravitational force, towards the planet’s center of mass.
 Rotor ride: Centripetal force = Gravitational force, towards the planet’s center of mass.
 In this case, Weight force = Friction force.
Turning on a horizontal road: Centripetal force = Friction acting between the tyres and the road.
When skidding: (mv^2)/r = μdmg.
So that it does not skid: (mv^2)/r < μsmg.
Banking on the road: Road banked at an angle θ. Centripetal force = Normal force x sinθ.
Angle proportional to speed.
Examples: cars, cycle velodrome, commercial airline pilots, highspeed trains.
Vertical circle with strings: Weight force and string tension must be taken into account.
At the top: Fc = Tdown + mg
To keep on moving: v^2 = gr.
At the bottom: Fc = Tup – mg.
Maximum tension on the bottom so
that it does not break:
Tbreak > (mv^2)/r + mg
Car on speed bump: Car loses contact when Centripetal force = Weight force, i.e. N = 0.
UNIFORM AND NONUNIFORM CIRCULAR MOTION
UNIFORM CIRCULAR MOTION
NONUNIFORM CIRCULAR MOTION
 Angular displacement behaves like vector, when its magnitude is very small. It follows laws of vector addition.
 Angular velocity and angular acceleration are axial vectors.
 Centripetal acceleration always directed towards the centre of the circular path and is always perpendicular to the instantaneous velocity of the particle.
 Circular motion is uniform if aT = rα = 0, that is angular velocity remains constant and radial acceleration is constant.
 When aT or α is present, angular velocity varies with time and net acceleration is
 If aT = 0 or α = 0, no work is done in circular motion.