MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Circular Motion Page 3

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Examples:

 (1) Consider a satellite performing U.C.M. around the earth. In this case, the centripetal force needed for the U.C.M. is provided by the gravitational force exerted on the satellite by the earth. (2) Consider an electron performing U.C.M. around the nucleus of an atom. In this case, the centripetal force needed for the U.C.M. is provided by the electrostatic force of attraction exerted on the electron by the nucleus.

Pseudo Force (Fictitious Force)

 A force is called a real force if it can be explained in terms of interaction between a body and its surroundings. Examples: Gravitational force, electromagnetic force, nuclear force. A force is called a pseudo force if it cannot be explained in terms of interaction between a body and its surroundings. A pseudo force arises due to acceleration of the reference frame. Examples: (1) When a car takes a turn passengers in the car experience a force in the radially outward direction. This is a pseudo force called the centrifugal (directed away from the centre) force. (2) A coin placed slightly away from the centre of a rotating gramophone disc, slips towards the edge of the disc due to the centrifugal force acting on it.

 Centrifugal Force: It is defined as the radial force directed away from the centre of the circle and acting on a body in a rotating frame of reference.

U.C.M. of a Car when the Road Surface is Horizontal [Expression for Limiting Safety Speed:]

Consider a car moving with a constant speed along a curved path of radius of curvature r, on the horizontal road. The centripetal force required for the U.C.M. is provided by the friction between the road surface and the tyres of the car. If v is the limiting safety speed of the car, then we have,

 m v2 = m m g r

Where m is the coefficient of friction between the road surface and the tyres of the car

Therefore,
v2 = m r g
Therefore,
v =
If
v > , the car will be thrown off the road, leading to an accident.

For uniform circular motion of a car (or any other vehicle) there must be a centripetal force acting on the car. If the road is horizontal, the only force which can act as the centripetal force is the force of friction between the tyres of the car and the road surface.

However, the force of friction is usually very small compared to the centripetal force required. Also, the force of friction cannot be assumed to be constant because it depends upon number of factors such as,

 (1) Nature of the surfaces in contact. (2) Presence of water or oil on the road.

Therefore, there is a danger that a fast moving vehicle may skid and be thrown away from the road. Also, friction causes wear and tear of the tyres. Hence, some arrangement is required to provide the centripetal force.

To ensure safety of a fast moving vehicle taking a turn, at the turn, the surface of the road is kept inclined to the horizontal at a suitable angle such that the outer edge of the road is at a higher level than the inner edge. This arrangement is called banking of a road.

When the road is banked, the necessary centripetal force for the circular motion is provided by the horizontal component of the normal reaction.

Angle of Banking

Consider a car of mass 'm' moving with a speed v along a curved banked road of radius r; q is the angle of banking:

The forces acting on the car are:

 (i) Its weight mg acting vertically downwards. (ii) The normal reaction R acting perpendicular to the surface of the road.

 R cos q = mg ...(1)

 R sin q = mv2 ...(2) r

 R sin q R cos q

=

 ( mv2 ) r mg

 \ tan q = v2 rg

 \ q = tan-1 ( v2 ) rg

 or v = Þ maximum speed.

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