Mechanics

V

Torque

For equilibrium, the horizontal components of the forces acting on an object must cancel one another, and so must the vertical components. This condition is necessary for equilibrium, but not sufficient. For example, if a person stands a book on a table and pushes the book equally hard with one hand in one direction and with the other hand in the other direction, the book will remain motionless if the person's hands are opposite each other. (The net result is that the book is being squeezed.) If, however, one hand is near the top of the book and the other hand near the bottom, a torque, or turning force, is produced, and the book will fall on its side. For equilibrium to exist it is also necessary that the sum of the torques about any axis be zero.

A torque, or moment of a force, is the product of the force and the perpendicular distance to an axis of rotation. When a force is applied to a heavy door to open it, the force is exerted perpendicularly to the door and at the greatest distance from the hinges. Thus, a maximum torque is created. If the door were shoved with the same force at a point halfway between handle and hinge, the torque would be only half of its previous magnitude. If the force were applied parallel to the door (that is, edge on), the torque would be zero. For an object to be in equilibrium, the clockwise torques about any axis must be cancelled by the anti-clockwise torques about that axis. It can be proved that if the torques cancel for any particular axis, they cancel for all axes.

VI

Newton's Three Laws of Motion

Newton's first law of motion states that if the vector sum of the forces acting on an object is zero, then the object will remain at rest or remain moving at constant velocity. If the force exerted on an object is zero, the object does not necessarily have zero velocity. Without any forces acting on it, including friction, an object in motion will continue to travel at constant velocity.

A

The Second Law

Newton's second law relates net force and acceleration. A net force on an object will accelerate it—that is, change its velocity. The acceleration will be proportional to the magnitude of the force and in the same direction as the force. The proportionality constant is the mass, m, of the object

F = ma

In the International System of Units (also known as SI, the acronym for “Système International”), acceleration, a, is measured in m/s². Mass is measured in kg; force, F, in newtons. A newton is defined as the force necessary to impart to a mass of 1 kg an acceleration of 1 m/s²; this force is roughly equal to the weight of a 100-g (3.5-oz) object at sea level.

A more massive object will require a greater force for a given acceleration than a less massive one. What is remarkable is that mass, which is a measure of the inertia of an object (its reluctance to change velocity), is also a measure of the gravitational attraction that the object exerts on other objects. It is surprising and profound that the inertial property and the gravitational property are determined by the same thing. Einstein made this one of the cornerstones of his general theory of relativity, which is the currently accepted theory of gravitation (see Relativity: Theory of General Relativity).

B

Friction

Friction acts like a force applied in the direction opposite to an object's velocity. For dry sliding friction, where no lubrication is present, the friction force is almost independent of velocity. The friction force also does not depend on the apparent area of contact between an object and the surface upon which it slides. The actual contact area—that is, the area where the microscopic bumps on the object and sliding surface are actually touching each other—is relatively small. As the object moves across the sliding surface, the tiny bumps on the object and sliding surface collide, and force is required to move the bumps past each other. The actual contact area depends on the perpendicular force between the object and sliding surface. Frequently this force is just the weight of the sliding object. If the object is pushed at an angle to the horizontal, however, the downward vertical component of the force will, in effect, be added to the weight of the object. The friction force is proportional to the total perpendicular force.

Where friction is present, Newton's second law can be expanded to

When an object moves through a liquid, however, the magnitude of the friction depends on the velocity. For most human-sized objects moving in water or air (at subsonic speeds), the resulting friction is proportional to the square of the speed. Newton's second law then becomes

The proportionality constant, k, is characteristic of the two materials that are moving past each other, and depends on the area of contact between the two surfaces and the degree of streamlining of the moving object.

C

The Third Law

Newton's third law of motion states that when an object exerts a force on another object, it experiences a force in return. The force that object one exerts on object two must be of the same magnitude as the force that object two exerts on object one but in the opposite direction. On a skating rink, for example, if a large adult gently pushes away a child, then in addition to the force the adult exerts on the child, the child exerts an equal but oppositely directed force on the adult. Because the mass of the adult is larger, however, the acceleration of the adult will be smaller.

Newton's third law also requires the conservation of momentum, the product of mass and velocity. For an isolated system, with no external forces acting on it, the momentum must remain constant. In the example of the adult and child on the skating rink, their initial velocities are zero, and thus the initial momentum of the system is zero. During the interaction, internal forces are at work between adult and child, but net external forces equal zero. Therefore, the momentum of the system must remain zero. After the adult has pushed the child away, the product of the large mass and small velocity of the adult must equal the product of the small mass and large velocity of the child. The momenta are equal in magnitude but opposite in direction, thus adding up to zero.

Another conserved quantity of great importance is angular (rotational) momentum. The angular momentum of a rotating object depends on its speed of rotation, its mass, and the distance of the mass from the axis. When a skater on (almost) frictionless ice spins faster and faster, angular momentum is conserved despite the increasing speed. At the start of the spin, the skater's arms are outstretched. Part of the skater's mass is therefore at a large radius. As the skater's arms are lowered, thus decreasing their distance from the axis of rotation, the rotational speed must increase in order to maintain constant angular momentum.