Force of Friction

Dynamics of Circular Motion

Force on the particle: For a mass m to have an acceleration, a force given by F = ma is always required. In uniform circular motion, acceleration is of magnitude v²/r and directed towards centre. Hence a force of magnitude mv²/r and directed towards centre is required to keep a particle in circular motion. This force (acting towards centre) is known as the centripetal force. After studying the following examples, you should carefully note that the centripetal force is not an extra force on a body. Whatever force is responsible for circular motion becomes the centripetal force.

For example, when a satellite revolves around the earth, the gravitational attraction of earth becomes the centripetal force for the circular motion of the satellite; when an electron revolves around the nucleus in an atom. The electrostatic attraction of nucleus becomes the centripetal force for the electron's circular motion; in case of a conical pendulum T sin q (component of tension) becomes the centripetal force.

Main steps for analysing forces in uniform circular motion: Take one axis along the radius of circle (i.e. in direction of acceleration) and other axis perpendicular to the radius. Resolve all the force into components.

Net forces along perpendicular axis = 0

Net force along radial axis (towards centre ) =

Main steps for analysing forces in non-uniform circular motion: After resolving all the forces along tangential and radial axes:

Net tangential force =

Net radial force

The most common example of non-uniform circular motion is the motion of particle in vertical circle. If a particle is revolved in a vertical circle with the help of a string, the forces are : tension (T) towards centre and weight (mg). In case of a particle moving along the outside surface of a circular track (or sphere), the forces are : normal reaction (N) away from the centre and weight (mg).

Work Done by a Constant Force

Consider a particle acted on by a force. In the simplest case the force F is constant and the motion takes place in a straight line in the direction of the force. In such a situation we define the work done by the force on the particle as the product of the magnitude of the force F and the distance d through which the particle moves. We write this as

However, the constant force acting on the particle may not act in the direction in which the particle moves. In this case we define the work done by the force on the particle as the product of the component of the force along the line of motion by the distance d the body moves along that line. In the figure a constant force F acts along the X-axis on a particle whose displacement is d along a direction making angle q with the X-axis. In this case

We can write the above equation either as ( F cos q) d or (F) (cos q). This suggests that the work can be calculated in two different ways : Either we multiply the magnitude of the displacement by the component of the force in the direction of the displacement or we multiply the magnitude of the force by the component of the displacement in the direction of the force. These two methods always give the same result.

Thus work is defined as a dot product of the force acting on a body and the displacement of the body under that force.

Unit of Work

The unit of work is the work done by a unit force in moving a body a unit distance in the direction of the force. In S.I. system, the unit of work is 1 newton-metre or 1 joule (J). The work done is independent of time.

So, 1 J = 1 N × 1 m

If d = 0. W = 0. No work is done in pushing a wall or no work is done when a man remains standing with the load on his head.
If θ 90° W = 0 No work is done in walking on a plane road against gravitational force because the direction of motion is perpendicular to the direction of gravitational force.
Similarly, no work is done by the centripetal force on a particle in circular motion. The speed and K.E. of the particle do not change.
A force is said to be conservative if the work done by it in moving a particle from one point to another point depends only on these points and does not depend on the path followed (i.e. W is zero for the closed path). For example, gravitational and electrostatic force are the conservative forces. If the work done depends upon the path and is not zero for closed path, the force is said to be non conservative, For example viscous and frictional forces are non-conservative forces.
Work done may be either positive or negative. If the particle on which the forces act has a component of motion opposite to the direction of the force, the work done by that force is negative. This corresponds to an obtuse angle between the force and the displacement vectors. Naturally, if the angle between the force and displacement vectors is acute, the work done by that focre is positive. For example, when a person lowers an object to the floor, the work done on the object by the upward force on his hand holding the object is negative. In this case θ= 180° for F points up and d points down.