21.
In physics,
the center
of mass of
a distribution of mass in
space is the unique point where the weighted relative position of
the distributed mass sums to zero. The distribution of mass is
balanced around the center of mass and the average of the weighted
position coordinates of the distributed mass defines its coordinates.
Calculations in mechanics are
often simplified when formulated with respect to the center of mass.
22.
23. Conservation of Angular Momentum Angular Momentum
O
bjects
executing motion around a point possess a quantity called angular
momentum.
This is an important physical quantity because all experimental
evidence indicates that angular momentum is rigorously conserved in
our Universe: it can be transferred, but it cannot be created or
destroyed. For the simple case of a small mass executing uniform
circular motion around a much larger mass (so that we can neglect the
effect of the center of mass) the amount of angular momentum takes a
simple form. As the adjacent figure illustrates the magnitude of the
angular momentum in this case is L
= mvr,
where L is
the angular momentum, m is
the mass of the small object, v is
the magnitude of its velocity, and r is
the separation between the objects.
Ice Skaters and Angular Momentum
This formula indicates one important physical consequence of angular momentum: because the above formula can be rearranged to give v = L/(mr) and L is a constant for an isolated system, the velocity v and the separation r are inversely correlated. Thus, conservation of angular momentum demands that a decrease in the separation r be accompanied by an increase in the velocity v, and vice versa. This important concept carries over to more complicated systems: generally, for rotating bodies, if their radii decrease they must spin faster in order to conserve angular momentum. This concept is familiar intuitively to the ice skater who spins faster when the arms are drawn in, and slower when the arms are extended; although most ice skaters don't think about it explictly, this method of spin control is nothing but an invocation of the law of angular momentum conservation.
24.
n physics, a force is said to do work if, when acting on a body, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the work done on the ball as it falls is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement).
Provide examples of work done by variable and constant forces Using Integration to Calculate the Work Done by Variable Forces
A force is said to do work when it acts on a body so that there is a displacement of the point of application in the direction of the force. Thus, a force does work when it results in movement.
The work done by a constant force of magnitude F on a point that moves a displacement Δx in the direction of the force is simply the product
W=F⋅Δx
In the case of a variable force, integration is necessary to calculate the work done. For example, let's consider work done by a spring. A horizontal spring exerts a force with magnitude
F=k⋅Δx
that is proportional to its displacement in the x direction from the spring's equilibrium position. For a variable force, one must add all the infinitesimally small contributions to the work done during infinitesimally small time intervals dt (or equivalently, in infinitely small length intervalsdx=vxdt). In other words, an integral must be evaluated.
W=∫t0F⋅vdt=∫t0kxvxdt=∫xxokxdx=12kΔx2
This is the work done by a spring exerting a variable force on a mass moving from position xo to x (from time 0 to time t). The work done in positive if the applied force is in the same direction as the direction of motion.