ChemPhys 173/273

Unit 11: Work, Energy and Power

Problem Set E

Overview:

Problem Set E contains 17 problems which test your ability to use the work-energy relationship to analyze a physical situation and solve for some unknown quantity. Most problems are complex, demanding both strong algebra skills and the ability to accurately read and interpret a verbal description of a physical situation. The set represents a classic case in which you will have to blend a sturdy conceptual understanding with strong math skills to be successful.

The Theory:

The fundamental principle behind this problem set is that the initial amount of total mechanical energy (TMEi) of a system is altered by the work which is done to it by non-conservative (or external) forces. The final amount of total mechanical energy (TMEf) possessed by the system is equivalent to the initial amount of energy plus the work done by these non-conservative forces.

TMEi + Wnc = TMEf

The energy possessed by a system is the sum of the kinetic energy and the potential energy. Thus the above equation can be re-arranged to the form of

KEi + PEi + Wnc = KEf + PEf

The work done to a system by non-conservative forces (Wnc) can be described as either positive work or negative work. Positive work is done on a system when the force doing the work acts in the direction of the motion of the object. Negative work is done when the force doing the work opposes the motion of the object. When a positive value for work is substituted into the work-energy equation above, the final amount of energy will be greater than the initial amount of energy; the system is said to have gained mechanical energy. When a negative value for work is substituted into the work-energy equation above, the final amount of energy will be less than the initial amount of energy; the system is said to have lost mechanical energy. There are occasions in which the only forces doing work are conservative forces (sometimes referred to as internal forces). In general, such forces include gravitational forces, elastic or spring forces, electrical forces and magnetic forces. When the only forces doing work are conservative forces, then the Wnc term in the equation above is zero. In such instances, the system is said to have conserved its mechanical energy.

The Practice:

In Problem Set C, you will have to carefully read the problem description and substitute values from it into the work-energy equation listed above. You will also have to make inferences about certain terms based on your conceptual understanding of kinetic and potential energy. For instance, if the object is initially on the ground, then you can infer that its PEi is 0 and that term can be canceled from the work-energy equation. In other instances, the height of the object is the same in the initial state as in the final state, so the PEi and the PEf terms are the same. As such, they can be mathematically canceled from each side of the equation. In other instances, the speed is constant during the motion, so the KEi and KEf terms are the same and can thus be mathematically canceled from each side of the equation.

If a term does not cancel out of the work-energy equation, then you will have to use given values to calculate the term. In such instances, the KE, PE and Wnc terms can be determined using their respective equations.

 KE = 0.5 • m • v2 PE = m • g • h Wnc = F • d • cos()

In some of the more difficult problems, stated values will have to be treated to convert to standard units. In other cases, the force will have to be calculated using Newton's laws or an inclined plane analysis.

To be successful on this problem set, you will need to be able to:

• give attention to the units of stated quantities and peform proper conversions.
• determine the value of a friction force as mu•Fnorm.
• determine the perpendicular component of gravity on an inclined plane (in order to determine the friction force on an incline): Fperp = m•g•cos().
• use trigonometry to relate the distance traveled along the length of an inclined plane to the inclined angle and the height of the plane at a given point.
• determine the power of a machine using P = F•v where v is either the average or the constant velocity.
• calculate kinetic and potential energies and work values.
• calculate percentages.
• use the work-energy equation.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Work | Sample Work Calculations | Potential Energy | Kinetic Energy | Mechanical Energy

Resolving Fgrav on an Inclined Plane

View Sample Problem Set.

 Problem Description Audio Link 1 Analysis of a child on a swing; the percent energy lost is determined from the initial height and the final speed. 2 Determination of the average water resistance force on a diver who starts from a known height and finishes a known distance below the water. 3 Determination of the distance traveled by a skater along ice with known friction coefficient and known initial speed. 4 Calculation of the final speed of a grocery cart if given the force acting upon it and its displacement. 5 Analysis of a car moving down the incline of a driveway; determination of the final speed if given the amunt of friction force and the initial height. 6 Determination of the amount of lost energy in a swinging child if given its initial height and the final speed. 7 Analysis of a sled along an incline plane; the distance moved upward along the incline must be detemined from the incline angle and the approach speed. 8 Analysis of a block along an incline plane; the distance moved upward along the incline must be detemined from the incline angle and the approach speed. 9 Determination of the distance moved by a skater with a known initial speed and a known friction coefficient. 10 Determination of the work required to acceleraate a plane upward from rest with a given acceleration and a given time. 11 Determination of the power required of a machine to pull a crate at a constant speed across a level surface of known friction coefficient. 12 Determination of the work done by friction on a bead moving along a curved path with known initial and final heights and speeds. 13 Analysis of the motion of a baseball being caught by a catcher; must determined the force exerted by the mitt on the ball over a known distance to stop a ball with known initial speed. 14 Determination of the force exerted upon a man while landing after stepping off a table of known height. 15 Determination of the final speed of an object being lifted by a rope with known tension through a known distance. 16 Calculation of the amount of work done on an athlete who is ascending along an incline with a constant speed a known distance in a known amount of time. 17 Complex analysis of a two-body system with one body moving along an incline plane and the other hanging over a pulley; must determine the change in kinetic energy of the body on the incline.

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