Centripetal Acceleration (Copy)

Understanding Centripetal Force and Centripetal Acceleration

• Circular Motion:
  • Occurs when an object moves in a circular path with a constant speed.
  • Even though speed is constant, velocity changes due to changing direction.
  • Any change in velocity implies acceleration — this is known as centripetal acceleration.
• Centripetal Acceleration:
  • Acceleration that acts towards the centre of the circular path.
  • It is responsible for changing the direction of velocity, not its magnitude.
  • Caused by a resultant force directed towards the centre (e.g., tension, gravity, friction, etc.).
  • It exists even when angular speed is constant, because direction of motion is continuously changing.
• Perpendicular Force:
  • In uniform circular motion, the force causing centripetal acceleration is always perpendicular to the instantaneous direction of motion (which is tangential).
  • This is why it changes direction but not the speed.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Centripetal Acceleration Equations

• Centripetal acceleration can be expressed in two equivalent ways:

Formula 1: a = v² / r

• a = centripetal acceleration (m/s²)
• v = linear (tangential) speed (m/s)
• r = radius of the circular path (m)
• Derived from the concept of constant change in direction of velocity vector.

Formula 2: a = rω²

• Derived using the relationship v = rω
• Substituting into a = v² / r:
  • a = (rω)² / r = r²ω² / r = rω²
• ω = angular speed (rad/s)
• Highlights that for a given radius, centripetal acceleration increases with the square of angular speed.

Derivation of a = rω² from a = v² / r

• Start with: v = rω
• Substitute into a = v² / r:
  • a = (rω)² / r = rω²

Forces Causing Circular Motion

• According to Newton’s 2nd Law:
  • F = ma, and for circular motion: F = m × a_centripetal
• Hence, centripetal force (F) has two equivalent expressions:

F = mv² / r

• Based on linear speed
• Useful when v is given

F = mrω²

• Based on angular speed
• Derived using v = rω → plug into first formula

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Key Concepts Recap

Nature of the Force Providing Centripetal Acceleration

• Important Clarification:
  • Centripetal force is not a new type of force.
  • It is the resultant of existing forces such as:
    • Tension (in a string)
    • Gravitational force (in planetary orbits)
    • Friction (in vehicle turning)
    • Normal reaction force
• Direction:
  • Always points inward towards the centre of the circular path.
  • Always perpendicular to velocity.
• If the force is removed:
  • The object will move tangentially to the circle (not spiral out) as per Newton’s First Law.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Examples of Centripetal Force in Action

• Car on a circular track:
  • Friction between tyres and road provides the centripetal force.
• Satellite orbiting Earth:
  • Gravitational force provides the required centripetal force.
• Object on a string:
  • Tension in the string provides centripetal force.
• Loop-the-loop roller coaster:
  • At the top of the loop, weight + normal force provide the centripetal force.

Problem Solving Examples

Q1: A 1.2 kg mass moves in a circle of radius 0.5 m at 3 m/s. What is the centripetal force?

• F = mv² / r = 1.2 × 3² / 0.5 = 1.2 × 9 / 0.5 = 21.6 N

Q2: A particle moves in a circle of radius 0.8 m with angular speed 5 rad/s. What is its centripetal acceleration?

• a = rω² = 0.8 × 5² = 0.8 × 25 = 20 m/s²

Q3: An object of mass 2 kg rotates at angular speed 4 rad/s in a circle of radius 0.6 m. What is the force acting on it?

• F = mrω² = 2 × 0.6 × 4² = 2 × 0.6 × 16 = 19.2 N

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Common Misconceptions

• Centripetal vs Centrifugal:
  • Centripetal is the real, inward force causing circular motion.
  • Centrifugal is a fictitious force felt in the rotating frame, directed outward, but it does not exist in an inertial frame.
• Object moving in a circle is not in equilibrium:
  • Though the speed is constant, the acceleration is not zero.
  • Hence, net force ≠ 0, and the object is not in equilibrium.
• Centripetal force is not a type of force:
  • It is the name given to the resultant force causing circular motion.
  • It must always be identified: is it tension? friction? gravity?

Units Recap

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course