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Centripetal Force

Acceleration of a body moving in a circle of radius R with uniform speed υis υ2/R directed towards the centre. According to second law, the force f providing this acceleration is
where m is the mass of the body. This directed towards the centre is called the centripetal force.
For a stone rotated in a circle by a string, the centripetal force is provided by the tension in the string.
The centripetal force for motion of a planet around the sun is the gravitational force on the planet due to the sun.
For a car taking a circular turn on a horizontal road, the centripetal force is the force of friction.
Let us discuss the circular motion of a car on a flat and banked road.

Motion of a Car on a Level Road

Three forces act on a car:
  • The weight of the car

  • Normal reaction, N

  • Frictional force, f

As there is no acceleration in the vertical direction.
N - mg = 0
N = mg
The centripetal force required for circular motion is along the surface of the road, and is provided by the component of the contact force between road and the car tyres along the surface. This by definition is the frictional force. It is the static friction that provides the centripetal acceleration. Static friction opposes the impending motion of the car moving away from the circle.
which is independent of the mass of the car.
The above equation shows that for a given value of μs and R, there is a maximum speed of circular motion of the car possible.

Motion of a Car on a Banked Road

We can reduce the contribution of friction to the circular motion of the car if the road is banked. Since there is no acceleration along the vertical direction, the net force along this direction must be zero. Hence,
N cos q = mg + f sin θ
The centripetal force is provided by the horizontal components of N and f.

Thus to obtain υmax, we put,

N cos θ=mg + μs N sin θ 

Substituting the value of N in the above equation,
The maximum possible speed of a car on a banked road is greater than that on a flat road.
μs = 0, then
At this speed, frictional force is not needed at all to provide the necessary centripetal force. Driving at this speed on a banked road will cause little wear and tear of the tyres. The same equation also tells us that for υ< υ0,the frictional force will be up the slope and that a car can be parked only if .

Solving Problems in Mechanics

In mechanics, many a time, the problems do not involve merely single body under the action of a single force or a number of forces. In certain problems, one may come across a system consisting of a number of bodies exerting forces on each other through various kinds of supports, connecting strings, etc. In addition, other forces such as the gravitational force, frictional force, etc may also be acting between bodies.

Under such circumstances, each body is to be considered separately. Equation of motion for each body is to be obtained, taking into account all the forces acting on it and then equating the net force acting on the body to its mass times the acceleration produced

A diagram for each body of the system depicting all the forces on the body by the remaining part of the system is called the free body diagram.
The equations of motion obtained for different bodies can be solved to find the unknown quantities.
To handle a typical problem in mechanics systematically, one should use the following steps;
  • Draw a diagram showing schematically the various parts of the assembly of bodies, the links, supports etc.
  • Choose a convenient part of the assembly as one system.
  • Draw a separate diagram which shows this system and all the forces on the system by the remaining part of the assembly. Include also the forces on the system by other agencies. Do not include the forces on the environment by the system. A diagram of this type is known as Free Body Diagram.
  • In a free body diagram, include information about forces that are either given or you are sure of. The rest should be treated as unknowns to be determined using laws of motion.
  • if necessary, follow the same procedure for another choice of the system. in doing so, employ the Newton's third law of motion. That is, if in the free-body diagram of A, the force on A due to B is shown as F, then in the free body diagram of B, the force on B due to A should be shown as -F.

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