For procedures, see your mechanical laboratory manual.

Title: Rolling on an Inclined plane.

Aim

To find the radius of gyration of a body rolling on an Inclined plane and to verify Newton's Laws in this case.

Apparatus

1. Stopwatch
2. Meter rule
3. Rolling bodies
4. Inclined plane
5. Wheel and Axle

Theory

An object placed on a tilted surface will often slide down the surface. The rate at which the object slides down the surface is dependent upon how tilted the surface is; the greater the tilt of the surface, the faster the rate at which the object will slide down it. In physics, a tilted surface is called an inclined plane. An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. Objects are known to accelerate down inclined planes because of an unbalanced force. Inclined planes are widely used to move heavy loads over vertical obstacles; examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.

Moving an object up an inclined plane requires less force than lifting it straight up, at a cost of an increase in the distance moved. The mechanical advantage of an inclined plane, the factor by which the force is reduced, is equal to the ratio of the length of the sloped surface to the height it spans. Due to the conservation of energy, the same amount of mechanical energy (work) is required to lift a given object by a given vertical distance, disregarding losses from friction, but the inclined plane allows the same work to be done with a smaller force exerted over a greater distance. The angle of friction, also sometimes called the angle of repose, is the maximum angle at which a load can rest motionless on an inclined plane due to friction, without sliding down. This angle is equal to the arctangent of the coefficient of static friction μs between the surfaces. The inclined plane reduces the effect of the force of gravity on an object on the inclined surface by changing the angle of application of the force component.

Newton’s Second Law provides the basic theory for this experiment. It states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. Also, the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration "a" of the object: F = ma; where F = net resultant force acting on the object, m = mass of the object and a = acceleration produced by the body due to the effect of forces.

The radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance of a point, from the axis of rotation at which, if the whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass. It is the distance from the center of mass of a body at which the whole mass could be concentrated without changing its moment of rotational inertia about an axis through the center of mass.

Applications

1. Escalators and slanted conveyor belts
2. Wheelchair ramps
3. In a funicular or cable railway a railroad car
4. Aircraft evacuation slides

References

1. Alfred Rudin, Phillip Choi, in The Elements of Polymer Science & Engineering (Third Edition), 2013
2. Radius of gyration, sciencedirect.com
3. Inclined plane, wikipedia.org
4. Inclined planes, physicsclassroom.com