Revision Notes: Circular Motion | Physics for JEE Main & Advanced PDF Download

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.

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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.

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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.

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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,