# Unit 6. Circular motion and gravitation: Circular motion Notes

 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

### 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, ac, and Fc 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 Fc N ### Defining Circular Motion Period T s Angular Velocity ω rad s-1 Time per revolution ### Vertical Circular Motion Top: Bottom: Fnet = Fc = FT + Fg Fnet = Fc = FT – Fg Top: Bottom:    Fnet = Fc = Fg – R Fnet = Fc = R – Fg

### 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.

• 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, high-speed 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 NON-UNIFORM CIRCULAR MOTION

#### UNIFORM CIRCULAR MOTION

An object moving in a circle with a constant speed is said to be in uniform circular motion. Example – Motion of the tip of the second hand of a clock.

ANGULAR DISPLACEMENT : Change in angular position is called angular displacement (dθ). ANGULAR VELOCITY : Rate of change of angular displacement is called angular velocity ω
i.e., Relation between linear velocity (v) and angular velocity (ω). ANGULAR ACCELERATION : Rate of change of angular velocity is called angular acceleration.
i.e., Relation between linear acceleration and angular acceleration. CENTRIPETAL ACCELERATION : Acceleration acting on a body moving in uniform circular motion is called centripetal acceleration. It arises due to the change in the direction of the velocity vector.
Magnitude of centripetal acceleration is   This acceleration is always directed radially towards the centre of the circle.

CENTRIPETAL FORCE : The force required to keep a body moving in uniform circular motion is called centripetal force. It is always directed radially inwards.

CENTRIFUGAL FORCE : Centrifugal force is a fictitious force which acts on a body in rotating (non-inertial frames) frame of reference.
Magnitude of the centrifugal force This force is always directed radially outwards and is also called coriolis force.

#### NON-UNIFORM CIRCULAR MOTION

An object moving in a circle with variable speed is said to be in non-uniform circular motion.
If the angular velocity varies with time, the object has two accelerations possessed by it, centripetal acceleration (ac) and Tangential acceleration (aT) and both perpendicular to each other. Net acceleration KEEP IN MEMORY
1. Angular displacement behaves like vector, when its magnitude is very small. It follows laws of vector addition.
2. Angular velocity and angular acceleration are axial vectors.
3. Centripetal acceleration always directed towards the centre of the circular path and is always perpendicular to the instantaneous velocity of the particle.
4. Circular motion is uniform if aT = rα = 0, that is angular velocity remains constant and radial acceleration is constant.
5. When aT or α is present, angular velocity varies with time and net acceleration is 6. If aT = 0 or α = 0, no work is done in circular motion.