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Tuesday, 28 February 2017

Law of Motion XVI



Centripetal Force : Force responsible for producing centripetal acceleration is known as centripetal force. Since centripetal acceleration is directed towards the center of the circular path the centripetal force is also directed towards the center of the circular path.
If a body is performing uniform circular motion with speed v and angular velocity ω on a circular path of radius r, then centripetal acceleration is given by
Net Acceleration of a Body Performing Non-Uniform Circular Motion :
When a body performs non-uniform circular motion its speed i.e. magnitude of instantaneous velocity varies with time due to which it experiences tangential acceleration aT along with the centripetal acceleration ac. Since both the accelerations act simultaneously on a body and are mutually perpendicular to each other, the resultant acceleration aR is given by their vector sum.
Physical Application of Centripetal Force
i) Case - 1
Circular motion of a stone tied to a string.
Centripetal force is provided by the tension of the string
Fc = mv2/r = T

ii) Case - 2
Circular motion of electron around the nucleus.
Centripetal force is provided by the electrostatic force of attraction between the positively charged nucleus and negatively charged electron
Fc = mv2/r = FE

iii) Case - 3
Circular motion of planets around sun or satellites around planet.
Centripetal force is provided by the gravitational force of attraction between the planet and sun
Fc = mv2/r = Fg

iv) Case - 4

Circular motion of vehicles on a horizontal road.

Centripetal force is provided by the static frictional force between the road and the tyre of the vehicle.
Fc = mv2/r = Fs

v) Case - 5
Circular motion of a block on rotating platform.
Centripetal force is provided by the static frictional force between the block and the platform.
Fc = mv2/r = Fs

vi) Case - 6
Circular motion of mud particles sticking to the wheels of the vehicle.
Centripetal force is provided by the adhesive force of attraction between the mud particles and the tyres of the vehicle.
Fc = mv2/r = Fadhesive
At very high speed when adhesive force is unable to provide necessary centripetal force, the mud particles fly off tangentially. In order to prevent the particles from staining our clothes, mud-guards are provided over the wheels of vehicle.
vii) Case - 7
Circular motion of a train on a horizontal track.
Centripetal force is provided by the horizontal component of the reaction force applied by the outer track on the inner projection of the outer wheels
Fc = mv2/r = FHorizontal
viii) Case - 8
Circular motion of a toy hanging from ceiling of vehicle.
Whenever car takes a turn, string holding the toy gets tilted outward such that the vertical component of the tension of string balances the weight of the body and the horizontal component of tension provides the necessary centripetal force.
ix) Case - 9
Conical pendulum.
Whenever bob of a pendulum moves on a horizontal circle it’s string generates a cone. Such a pendulum is known as conical pendulum. The vertical component of the tension of the string balances the weight of the body and the horizontal component of tension provides the necessary centripetal force.
x) Case - 10
Well of death.

In the well of death, the rider tries to push the wall due to its tangential velocity in the outward direction due to which wall applies normal reaction in the inward direction. The vertical component of the normal reaction balances the weight of the body and its horizontal component provides the necessary centripetal force.
While taking a turn aero-plane tilts slightly inwards due to which it’s pressure force also gets tilted inwards due to which it’s pressure force also gets tilted inwards such that it’s vertical component balances the weight of the body and the horizontal component provides the necessary centripetal force.

xi) Case - 11
Banking of Roads
In case of horizontal road necessary centripetal force mv2/r is provided by static frictional force. When heavy vehicles move with high speed on a sharp turn (small radius) then all the factors contribute to huge centripetal force which if provided by the static frictional force may result in the fatal accident.
To prevent this roads are banked by lifting their outer edge. Due to this, normal reaction of road on the vehicle gets tilted inwards such that it’s vertical component balances the weight of the body and the horizontal component provides the necessary centripetal force.
Therefore, Tan θ = v2 / rg

xii) Case - 12
Bending of Cyclist
In case of a cyclist moving on a horizontal circular track necessary centripetal force is provided by static frictional force acting parallel along the base. As this frictional force is not passing from the center of mass of the system it tends to rotate the cycle along with the cyclist and make it fall outward of the center of the circular path.
To prevent himself from falling, the cyclist leans the cycle inwards towards the center of the circle due to which the normal reaction of the surface of road on the cycle also leans inward such that that its vertical component balances the weight of the body and the horizontal component provides the necessary centripetal force.

xiii) Case - 13

Motion of a Ball in a Bowl
When the bowl rotates with some angular velocity ω. The vertical component of the normal reaction of the bowl on the ball balances the weight of the body and its horizontal component provides the necessary centripetal force.

xiv) Case - 14

Motion of a train on the banked tracks.
At the turns tracks are banked by slightly elevating the outer tracks with respect to the inner ones. This slightly tilts the train inwards towards the center of the circular path due to which the normal reaction of the tracks on the train also gets slightly tilted inwards such that the vertical component of the normal reaction balances the weight of the train and it’s horizontal component provides the necessary centripetal force.

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