Laws of circular motion ( Centripetal Acceleration , Tangential linear Velocity & Centripetal Force )

Centripetal Acceleration

Changing the direction of velocity leads to the existence of acceleration called the centripetal acceleration ( a ) which is the acceleration acquired by an object moving in a circular path due to a continuous change in the direction of its velocity .

When a force ( F ) acts on a body of mass ( m ) moving at speed ( v ) in a circular path of radius ( r ) normally to the direction of its motion :

The magnitude of velocity ( v ) remains constant along its path , The direction of velocity changes from one point to another on its path .

Finding the value of the centripetal acceleration

If a body moves from point ( A ) to point ( B ) , the velocity ( v ) changes in direction but maintains a constant magnitude .

a = v² / r

Laws of circular motion

Laws of circular motion

Factors affecting the centripetal acceleration ( a ) :

The tangential velocity : Centripetal acceleration is directly proportional to square of the tangential velocity at constant radius of the circular path .

Slope = a / v² = 1 / r

The radius of circular path : Centripetal acceleration is inversely proportional to the radius of the circular path at constant tangential velocity .

Slope = a r = v²

Tangential linear Velocity

Finding the tangential linear velocity

If the body complete one circular revolution in an interval of time ( T ) which is called the periodic time , Periodic time is the time taken by the body to make one complete revolution .

The tangential linear velocity = Distance ( circle circumference ) / Periodic time

v = 2 πr / T

Factors affecting the tangential linear velocity ( v ) :

The radius of circular path : Tangential velocity is directly proportional to the radius of the circular path at constant periodic time .

Slope = v / r = 2 π / T

The periodic time : Tangential velocity is inversely proportional to the periodic time at constant radius of circular path .

Slope = v T = 2 π r

v = 2 π r / T

Calculating the angular velocity :

If a body moves at tangential velocity ( v ) in a circle of radius ( r ) from point ( A ) to point ( B ) , covering a distance ( Δ l ) corresponding to an angle ( Δθ ) , during time interval ( Δ t ) .

Then the value ( Δθ / Δ t ) is known as the angular velocity ( ω ) .

 ω = Δθ / Δ t

It is known that the value of the angle in radian equals the ratio between the arc length and the radius of the path .

Δθ = Δl / r 

ω = ( Δl / Δ t ) × ( 1 / r ) = v / r      ∴  v = ω r

Tangential ( linear ) velocity = Angular velocity × Radius of the path

Since v = 2 π r / T   , then  :  ω r = 2 π r / T

∴ ω = 2 π / T

The Centripetal Force

Finding the Centripetal Force ( F )

According to Newton’s Second Law , the force is given by the relation : 

F = m a   ,  a = v² / r

F = m a = m v² / r

Factors affecting the centripetal force ( F ) :

The object mass : Centripetal force is directly proportional to the object mass at constant tangential velocity and radius of circular path .

Slope = F / m = v² / r

The tangential velocity : Centripetal force is directly proportional to square of the tangential velocity at constant radius of circular path and object mass .

Slope = F / v² = m / r

F = m v² / r

The radius of circular path : Centripetal force is inversely proportional to the radius of the circular path at constant tangential velocity and object mass .

Slope = F r = m v²

Important applications

Designing curved roads :

It is necessary to calculate the centripetal force when designing the curved roads and railways in order to keep cars and trains moving along this curved path without skidding out .

If a car moves in a curved slippery road , the frictional forces may not be sufficient to turn the car around the curve , So , the car skids out the road .

Engineers define certain velocities for vehicles when moving in curves , If the vehicle velocity exceeds the predetermined velocity , the vehicle needs more more centripetal force to be kept in this curved path where F α v² .

It is forbidden for trucks and trailers to move on some dangerous curves , As the vehicle mass increases , it needs more centripetal force where F α m .

Slowing down in dangerous curves is a must to avoid accidents , As the radius of curve decreases the car needs more centripetal force to turn around , where F α 1/r .

When moving a bucket half filled with water in a vertical circular path at sufficient speed , the water does not spill out from the opening of the bucket because :

The centripetal force acting on water is normal to the direction of its motion , This force changes the direction of velocity without changing its magnitude to keep water inside the bucket rotating in a circular path .

We can make benefit of skidding objects out the circular path when the centripetal force is not sufficient to keep them rotating in the circular path :

  1. Making candy floss .
  2. Rotating barrels in amusement park .
  3. Drying cloths in automatic washing machines : water droplets adhere to clothes by certain forces , when the dryer rotates at great velocity , these adhesive forces will not be sufficient to hold these droplets , They separate from the cloths and move tangential to the circular path .

On using electric sharpener , the glowing metal splints blow in straight lines at tangent velocities .

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