Revision Notes: Circular Motion

Table of Contents
1. Introduction to Circular Motion
2. Types of Circular Motion
3. Variables in Circular Motion
4. Centripetal Acceleration
5. Kinematical Equations in Circular Motion
6. Centrifugal Force
7. Turning at Roads
8. Motion in a Vertical Circle
9. Non-uniform Horizontal Circular Motion
10. Conical Pendulum
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Introduction to Circular Motion

• Circular motion involves the movement of an object along a circular path.
• The object follows a curved trajectory rather than a straight line.
• It is characterized by a continuous change in direction as the object revolves around a fixed point or axis.
• Circular motion is commonly observed in various natural phenomena and man-made systems.
• The centripetal force is responsible for keeping the object in its circular path, preventing it from moving in a straight line.
• Examples of circular motion include the Earth revolving around the sun and a car navigating a curved road.

Types of Circular Motion

Uniform Circular Motion:

• In this type of circular motion, the magnitude of the particle's velocity remains constant.
• The object moves along a circular path with a consistent speed.

Non-uniform Circular Motion:

• In non-uniform circular motion, the magnitude of the body's velocity is not constant.
• The speed of the object changes as it traverses its circular trajectory.

Spinning Motion:

• A special category of circular motion occurs when an object rotates around its own axis.
• This type of motion, known as spinning, involves the object revolving around itself.

Variables in Circular Motion

Angular Displacement

• Definition: Angle subtended by the position vector at the center of the circular path.
• Formula:
• Units: Radian

Angular Velocity

• Definition: Time rate of change of angular displacement (Δθ). 
• Formula:
• Properties: Vector quantity, units in rad/s.
• Relation with Linear Velocity: v = rω

Angular Acceleration:

• Definition: Time rate of change of angular velocity
• Units: Rad/s², Dimensional Formula: [T^-2].
• Relation with Linear Acceleration: a=rα, where r is the radius.

Centripetal Acceleration

Centripetal Acceleration:

• Acceleration acting on a body in circular motion.
• Always directed towards the center of the circular path.
• Also known as radial acceleration, acting along the radius of the circle.
• Unit: m/s².
• Vector quantity.

Centripetal Force:

• Force compelling a body to move in a circular path.
• Directed along the radius of the circle towards its center.
• Not a new force; any existing force can act as a centripetal force.
  
• Formula: where m is mass, v is linear velocity, r is radius, and ac is centripetal acceleration.

Work Done by Centripetal Force:

• Zero, as the force and displacement are at right angles to each other.

Examples of Centripetal Force:

• Various incidents involve centripetal force:
  • Car Turning:
    • Tires exert centripetal force for the car's circular motion.
  • Satellite Orbiting Earth:
    • Gravitational force acts as centripetal force.
  • Whirling a Ball on a String:
    • Tension in the string provides centripetal force.

Kinematical Equations in Circular Motion

• Kinematic Equations in Circular Motion:

  • Describe relationships between various variables for an object undergoing circular motion.
    
  • Variables involved:
    • ω₀: Initial angular velocity.
    • ω: Final angular velocity.
    • α: Angular acceleration.
    • θ: Angular displacement.
    • t: Time.

• Key Variables:

  • ω₀: Represents the initial angular velocity of the object.
  • ω: Denotes the final angular velocity achieved during circular motion.
  • α: Signifies the angular acceleration experienced by the object.
  • θ: Indicates the angular displacement covered by the object.
  • t: Represents the time taken for the circular motion.

• Purpose:

  • These equations facilitate the calculation of various parameters involved in circular motion.

• Equations:

  • The kinematic equations in circular motion relate these variables and are essential for dynamic analysis.
    • Example: ω=ω₀+α⋅t
    • Example: θ=ω₀⋅t+1/2⋅α⋅t²

• Application:

  • Widely used in physics to understand and predict the motion of objects undergoing circular paths.

Centrifugal Force

Centrifugal Force:

• Equal and opposite to centripetal force in circular motion.
• Occurs when the centripetal force ceases to exist.
  
• Forces balance each other out in circular motion.

Motion Characteristics:

• Body moves only along a straight line under the influence of centrifugal force.
• Manifests when there is no centripetal force maintaining circular motion.

Frame of Reference:

• Centrifugal force does not act on the body in an inertial frame.
• Arises as a pseudo force in non-inertial frames.
• Important to consider in such frames for accurate analysis.

Balancing Forces:

• In circular motion, centripetal force pulls inward, and centrifugal force pushes outward, balancing each other.
• This balance allows the body to follow a curved path.

Considerations:

• Understanding centrifugal force is crucial when analyzing motion in non-inertial frames, as it plays a role in apparent forces in those situations.

Turning at Roads

• Frictional Centripetal Force:

  • Coefficient of friction (μ_s) between road and tires crucial for safe turning.
  • Centripetal force obtained solely from tire-road friction.
    
  • Relationship:  where v is velocity, and r is the radius of the circular path.

• Banked Roads for Centripetal Force:

  • Centripetal force solely from road banking.
  • Safe speed (v) for the turn determined by
  • If  the vehicle moves inward, decreasing radius; if , it moves outward, increasing radius.

• Combined Centripetal Force Sources:

  • In reality, centripetal force is a combination of friction and road banking.
  • Maximum safe speed significantly greater than the optimum speed on a banked road.

• Maximum Safe Speed Calculation:

  • Combining friction and banking, the maximum safe speed determined by the interplay of these forces.
    

• Cyclist Taking a Turn:

  • Cyclist inclines at an angle (θ) for safer turns.
  • Relationship:  where v is speed, r is the radius, and g is acceleration due to gravity.

• Safety Measures:

  • Slowing down and inclining on a larger radius crucial for cyclist safety during turns.

Motion in a Vertical Circle

(i) Minimum value of velocity at the highest point is √gr
(ii) The minimum velocity at the bottom required to complete the circle v_A = √5gr

(iii) Velocity of the body when string is in horizontal position v_B = √3gr
(iv) Tension in the string

• At the top T_c = 0,
• At the bottom T_A = 6 mg
• When string is horizontal T_B = 3 mg

(v) When a vehicle is moving over a convex bridge, then at the maximum height, reaction (N₁) is N₁ = mg - (mv²/r)
(vi) When a vehicle is moving over a concave bridge, then at the lowest point, reaction (N₂) is N₂ = mg + (mv²/r)

(vii) When a car takes a turn, sometimes it overturns. During the overturning, it is the inner wheel which leaves the ground first.
(viii) A driver sees a child in front of him during driving a car, then it, better to apply brake suddenly rather than taking a sharp turn to avoid an accident.

Non-uniform Horizontal Circular Motion

In non-uniform horizontal circular motion, the magnitude of the velocity of the body changes with time.

In this condition, centripetal (radial) acceleration (a_R) acts towards centre and a tangential acceleration (a_T) acts towards tangent. Both acceleration acts perpendicular to each other.

Resultant acceleration

where, α is angular acceleration, r = radius and a = velocity.

Conical Pendulum

Time period of conical pendulum,