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Article Outline
For equilibrium, all the horizontal components of the force must cancel one another, and all the vertical components must cancel one another as well. This condition is necessary for equilibrium, but not sufficient. For example, if a person stands a book up on a table and pushes on 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 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 is the product of a force and the perpendicular distance to a turning axis. 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 canceled by the counterclockwise torques about that axis. Therefore, one could prove that if the torques cancel for any particular axis, they cancel for all axes.
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.
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. A massive object will require a greater force for a given acceleration than a small, light object. What is remarkable is that mass, which is a measure of the inertia of an object (inertia is 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. The implication of this phenomenon is that it is impossible to distinguish at a point whether the point is in a gravitational field or in an accelerated frame of reference. Einstein made this one of the cornerstones of his general theory of relativity, which is the currently accepted theory of gravitation.
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. Also, the friction force 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, add to the weight of the object. The friction force is proportional to the total perpendicular force. Where friction is present, Newton’s second law
is expanded to Newton’s third law of motion states that an
object experiences a force because it is interacting with some other
object. The force that object 1 exerts on object 2 must be of the same
magnitude but in the opposite direction as the force that object 2 exerts
on object 1. If, for example, a large adult gently shoves away a child on
a skating rink, in addition to the force the adult imparts on the child,
the child imparts 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,
or 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 pushes 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 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 standing on a friction-free point 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 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
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© 2007 Microsoft
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