# 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**

## 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 :**

**Making candy floss .****Rotating barrels in amusement park .****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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