PLANAR MOTION OF RIGID BODIES

Planar Motion of Rigid Bodies



Planar Motion of Rigid Bodies

Introduction

One of the most fundamental physical laws governing the behavior of objects subject to forces is Newton's First Law: the velocity of an object subject to a net force of zero will remain constant. This means that if an object is initially at rest (velocity of zero), and no net force acts on this object, it will remain at rest. Consider as an example a person standing on a table. We all know that gravity exerts a downward force on the person (called the weight) as a result of the person's mass. With respect to Newton's First Law, how can this person remain at rest? The answer lies in the statement that the net force must be zero. Another force, equal and opposite to the gravitational force must be acting on the person so that their sum, the net force, will be zero. This force, which the table exerts on the person, is called the normal force and is the force exerted by a rigid object which prohibits penetration of the object. This force is always equal and opposite to the opposing force and perpendicular to the surface of the object in such a way that the sum of the normal and opposing forces is always zero. If it were not zero, the person in our example would accelerate through the table and hit the floor or in the opposite sense be repelled off the surface of the table and hit the ceiling. Now we will consider the case of an object which is placed on an inclined plane, a flat (planar) surface which is at some angle q with respect to level. (Featherstone 2008, pp. 10-19).

What are the forces at work on this object? If we neglect the friction force, the only two forces are the weight of the object caused by the gravitational force and the normal force exerted by the plane on the object. (Storch 1983, pp. 25-28).

On the left, the plane is shown with an inclination q . The force of the object's weight is represented by the vector W, and the normal force by the vector N. As you can see from the diagram on the right, the addition of W and N gives a resultant force W , which is in a direction parallel to the plane. In this sense, N can be considered as canceling the component of W perpendicular to the plane. W is the remaining component which is parallel to the plane. (Trinkle 1993, pp. 35-36).

Rigid body (ring, cylinder, sphere), who's radius of gyration is known, rolls down an inclined plane without slipping. Find velocity as a function of height. m - mass of the body. y0 - initial height, g - gravitational acceleration, v - speed. I - moment of inertia, K - radius of gyration. T - torque, a - linear acceleration, alfa - angular acceleration, w - angular ...