Chapter4

Energy Methods

1.
Energy methods greatly simplify the solution of problems involving $\rule{1.0in}{.01in}$ that depend on an object's $\rule{1.0in}{.01in}$ such as $\rule{1.4in}{.01in}$ $\rule{1.0in}{.01in}$ or $\rule{1.0in}{.01in}$ exerted by $\rule{1.0in}{.01in}$.

Principle of Work and Energy

1.
Write the mathematical expression for the principle of work and energy.





where





2.
U is the $\rule{1.0in}{.01in}$ and the term $\frac{1}{2}mv^2$ is called the $\rule{1.0in}{.01in}$ $\rule{1.0in}{.01in}$.


3.
Describe the principle of work and energy in words.






4.
The principle of work and energy relates changes in position to changes in velocity. However, you cannot use it to obtain other variables such as $\rule{1.0in}{.01in}$ or acceleration.

5.
To use the principle of work and energy the forces must be known as functions of $\rule{1.0in}{.01in}$.

6.
The principle of work and energy may be derived from Newton's second law which is stated in terms of velocity and time. The derivation requires that we dot both sides of Newton's second law with the velocity vector $\mbox{$\bar{{\bf v}}$}$ and rewrite the resulting expression in terms of the magnitude of the velocity and $\rule{1.0in}{.01in}$.


Work and Power

Evaluating the Work

1.
Sketch Figure 4.1.












2.
Write the expression for the work U in terms of the arc length s.





3.
Suppose the tangential force is constant. Write the simplification of the above expression.





Work Done by Various Forces

Weight

4.
Sketch Figure 4.6(a).












5.
Write the expression for the work done by the weight of an object as it moves between (x1,y1,z1) and (x2,y2,z2).



6.
On a separate piece of paper derive this expression as is done in the text.


7.
Sketch Figure 4.7.












8.
Write the expression for the work done by the weight of an object taking into account its variation with the distance from the earth.



Springs

9.
Sketch Figure 4.8.












10.
The expression for the work done on an object by a spring attached to a fixed support is



where S1 and S2 are the values of the $\rule{1.0in}{.01in}$ at the initial and final positions.


Power

11.
Write the definition of power (in words).





12.
Write an expression for power P in terms of force and velocity.



13.
Write an expression for power P in terms of kinetic energy.



14.
Write the expressions for the average power $P_{\rm av}$ transferred to or from an object during an interval of time in terms of both its kinetic energy and the work done.






15.
Recognize from the first expression that the average power is a measure of how fast the kinetic energy of an object is increased.

Conservation of Energy

1.
Write the fundamental relationship the potential energy V must satisfy in terms of the sum of the forces $\sum\mbox{$\bar{\bf F}$}$.



2.
Assume the forces satisfy the above relationship. Write the work done by the forces in terms of the potential energy as an object moves from position 1 to position 2.



3.
Write the conservation of energy expression in terms of V1 and V2.



4.
This principle may also be written as

\begin{displaymath} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= {\rm constant}.\end{displaymath}

5.
If a potential energy exists for a given force, the force is said to be $\rule{1.2in}{.01in}$.

6.
If all of the forces that do work on a system are conservative, the system is said to be $\rule{1.0in}{.01in}$ and the conservation of energy can be used instead of the principle of work and energy. These two approaches     are     are not     equivalent. (Circle the correct answer.)


Conservative Forces

1.
Friction forces     are     are not     conservative. (Circle the correct answer.) Potential Energies of Various Forces

2.
Below, write the equation describing the potential energy V of an object of mass m at a height y measured from a reference level or $\rule{1.0in}{.01in}$.



3.
Write the equation describing the potential energy V of an object of mass m at a distance r from the center of the earth, taking into account the variation of the weight with distance from the earth.



4.
Write the equation describing the potential energy V of a linear spring of spring constant k and stretch S.



Relationships Between Force and Potential Energy

5.
Write the relationship between the force and potential energy using cartesian coordinates.





where $\nabla V$ is the $\rule{1.0in}{.01in}$ of V.


Chapter Summary

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