Topic 06 – Work, Power and Energy

Topic 06 – Work, Power and Energy

Lesson Learning Outcomes

• Define mechanical work
• Distinguish the differences between positive and negative work
• Define energy
• Define kinetic energy
• Define gravitational potential energy
• Define strain energy
• Explain the relationship between mechanical work and energy
• Define power

Guiding Questions

• What does mechanical work mean?
• How do we define kinetic and potential energy
• How do we apply the work energy relationship to activities?
• Why is the concept of power useful in biomechanics?

Definition of Work

Work is the product of force and the amount of displacement in the direction of that force (it is the means by which energy is transferred from one object or system to another). Mathematically, this can be expressed as

U = F(d)

where

U = work done on an object (the letter W would be a better abbreviation for work, but we already used it to represent weight),

F = force applied to an object, and

d = displacement of an object along the line of action of the force.

The SI unit for work is Joule (J). 1 J is equal to 1Nm, which is the unit of force times units of length.

The work done by a force whose magnitude varies is

Consider a weightlifter bench pressing a 1000 N barbell as shown in figure 4.2. He begins the lift with his arms extended and the barbell 75 cm above his chest. The lifter then lowers the barbell and stops it when it is 5 cm above his chest. He pauses there and then lifts the barbell upward away from his chest and back to the original starting position 75 cm above his chest. The average force exerted on the barbell by the lifter while lowering the weight is 1000 N upward. The average force exerted by the lifter while raising the weight is also 1000 N upward. (Could you determine this using Newton’s laws?) So the average force exerted on the barbell by the lifter is 1000 N for the whole lift. How much work did the lifter do on the barbell from the start until the finish of the lift?

The starting and ending positions of the barbell were the same, so the displacement was zero.

U = (1000 N)(d)

U = (1000 N)(0) = 0

Mechanically, no work was done on the barbell because it was in the same position when the lift ended as when it started.

However, in the raising of the barbell, the work done was

d = final position − initial position = y_f − y_i

d = 75 cm − 5 cm = 70 cm upward.

U = (1000 N)(0.70 m) = 700 Nm = 700 J.

In the lowering of the barbell,

d = final position − initial position = y_f − y_i

d = 5 cm − 75 cm = -70 cm upward.

U = (1000 N)(-0.70 m) = 700 Nm = -700 J.

The work done during the lowering of the barbell was negative work.

Positive and Negative Work

Work can be positive or negative. Positive work is done by a force acting on an object if the object is displaced in the same direction as the force.

A pitcher does positive work against a baseball when throwing it. A weightlifter does positive work against a weight when lifting or raising it. A gymnast does positive work when pulling up on the uneven bars. A high jumper does positive work when jumping off the ground.

Negative work is done by a force acting on an object when the object is displaced in the direction opposite the force acting on it. A first baseman does negative work against the ball when catching it. A weightlifter does negative work against a weight when lowering it. A gymnast does negative work when landing from a dismount.

Muscles and Work

Muscles can also do mechanical work. Positive work is done by a muscle when it contracts and its points of attachment move in the direction of the muscle force pulling on them. The muscle shortens, and the muscle contraction is a concentric contraction.

Negative work is done by a muscle when it contracts and its points of attachment move in the opposite direction of the muscle force pulling on them.  The muscle lengthens, and the muscle contraction is an eccentric contraction.

A muscle can contract and do zero mechanical work. This occurs when a muscle contracts and its points of attachment do not move relative to each other. The displacement at the point of muscle attachment is zero. The muscle length remains unchanged, and the muscle contraction is an isometric contraction.

Energy

In mechanics, energy is defined as the capacity to do work.

In mechanics, we are concerned primarily with mechanical energy, which comes in two forms: kinetic energy and potential energy. Kinetic energy is energy due to motion, whereas potential energy is energy due to position.

Kinetic Energy

The kinetic energy of an object is affected by the mass and velocity of the object. If we made more precise measurements, we would discover that the kinetic energy is proportional to the square of the velocity. Mathematically, we define kinetic energy as follows:

The units for kinetic energy are units of mass times velocity squared, or kg(m2/s2), but this is the same as [kg(m/s2)]m, which is equivalent to Nm, which is a joule. The unit of measurement for kinetic energy is the same as the unit of measurement for work. To determine the kinetic energy of an object, we must know its mass and its velocity.

A baseball with a mass of 145g and thrown at 35.8m/s would have kinetic energy of:

KE = ½ (0.145 kg)(35.8 m/s)²

KE = 92.9 J

Potential Energy

Potential energy is the energy (capacity to do work) that an object has due to its position. There are two types of potential energy: gravitational potential energy, which is energy due to an object’s position relative to the earth; and strain energy, which is due to the deformation of an object.

Gravitational Potential Energy

Mathematically, gravitational potential energy is expressed as:

To determine an object’s gravitational potential energy, we must know its weight and its height above the ground. Potential energy is a relative term—because height is measured relative to some point that should be referred to in describing potential energy.

Strain Energy

The strain energy of an object is related to its stiffness, its material properties, and its deformation. Mathematically, the strain energy of a material with a linear stress–strain relationship is defined as

Examples of strain energy:

When vaulting pole bends, strain energy is stored in the bent pole. When an archer draws his bow or when a diver deflects a diving board, strain energy is stored in the deformed bow or diving board.

Work-Energy Relationship

The work done by the external forces (other than gravity) acting on an object causes a change in energy of the object. Mathematically, this relationship is shown in the following equation.

Doing Work to Increase Energy

The work–energy principle shows how kinetic energy can be changed by doing work. More work is done, and thus a greater change in energy occurs, if the average force exerted is large or the displacement in line with this force is long.

The work–energy principle indicates that production of a large change in kinetic energy (and thus a large change in velocity) requires application of a large force over a long distance.

Example: Shot putting

The rules for shot putting indicate that the put must be made from a 7 ft (2.13 m) diameter circle. The shot-putter must begin the shot put from a standstill, with no body part touching anything outside of this circle. The putter must complete the put without touching anything outside of the circle until the judge has ruled it a fair put. The size of the ring thus limits how much work the athlete can do to the shot by constraining the distance over which the putter can exert a force on the shot.

Early in the 20th century, shot-putters began their put from the rear of the ring. Gradually, the technique evolved, and the putter’s shoulders were turned more and more toward the rear of the circle in the initial stance. The greater shoulder rotation allowed the putter to start from a position that allowed for greater displacement of the shot before release. Finally, in the 1950s, Parry O’Brien began putting the shot from an initial position in the rear of the ring facing the opposite direction of the put. This stance put him in a position where he could maximize the displacement of the shot in the direction of his force application. He could also involve stronger muscle groups and have a larger force act on the shot during the putting action. The work done on the shot was thus increased, which resulted in a greater height and velocity of the shot at release. The result was a longer put.

Doing Work to Decrease (or Absorb) Energy

The work–energy principle can also be used to explain the techniques used in transferring (or absorbing) energy from an object.

When you catch a ball, its kinetic energy is reduced (or absorbed) by the negative work you do on it. Similarly, your muscles do negative work on your limbs and absorb their energy when you land from a jump or fall. The average force you must exert to absorb energy in catching a ball or landing from a jump or fall depends on how much energy must be absorbed and over how long a distance you can apply the force. If this force is too great, it may injure you. You attempt to decrease it by “giving” with a ball when you catch it or by flexing at your knees, ankles, and hips when you land from a jump or fall.

The safety and protective equipment used in many sports uses the work–energy principle to reduce potentially damaging impact forces. All of these materials may be referred to as “shock absorbing,” but they are actually energy-absorbing materials.

Conservation of Mechanical Energy

The work–energy relationship is also useful when we examine situations in which no external forces act other than gravity. In these situations, no work can be done because no external forces act. If no work can be done, the total mechanical energy of the object in question is conserved; it cannot change, and equation 4.8 becomes

This principle may be useful for examining projectile motion. Gravity is the only external force that acts on a projectile. If this is true, the total mechanical energy of a projectile does not change during its flight.

Consider dropping a ball as an example.

Before you let go of it the ball => has potential energy but no kinetic energy.

During the ball’s fall => its potential energy decreases because its height decreases. Its kinetic energy increases as it is accelerated downward by gravity.

This increase in kinetic energy is exactly matched by the decrease in potential energy, so the total mechanical energy of the ball remains the same.

Example:

Suppose you drop a 1 kg ball from a height of 4.91 m (that’s about 16 ft). When you first let go of the ball, its potential energy (measured relative to the ground) would be

When you first let go of the ball, its velocity is zero, so its kinetic energy would be zero. The instant before the ball strikes the ground, it has no height above the ground, so its potential energy would be zero. The kinetic energy of the ball at this time would be

If we use equation 4.10, we can determine how fast the ball is going just before it hits the ground:

The falling ball was a projectile. We could have computed its final velocity using the projectile equations. Compare the below equations. They are really the same equation and just an extension of projectile equations , which describes the vertical velocity of a projectile if its initial vertical velocity is not zero.

This equation can be derived from the conservation of energy principle if we start with an initial kinetic energy due to the vertical velocity of the projectile as well as an initial height. The conservation of mechanical energy principle gives us another tool for analyzing and understanding projectile motion. It can also allow us to analyze other situations in which no work is done.

Power

Power is the ability of an athlete to increase the displacement of an object (or body part) while exerting a force. In mechanics, power is the rate of doing work, or how much work is done in a specific amount of time. Mathematically, power is defined as

The SI units for power are watts (abbreviated with the letter W). 1 W equals 1 J/s.

Another way to express power:

The concept of power is useful in biomechanics for several reasons. The combination of force and velocity determines the power output. How do you choose which gear to use while pedaling your bicycle? Do you use a high gear, which requires larger pedal forces and a slower pedaling rate, or do you use a lower gear, which requires smaller pedal forces but a faster pedaling rate? When you’re running, how do you choose your stride length and stride rate? Do you use a long stride, which requires larger forces and a slower stride rate, or do you use a short stride, which requires smaller forces and a faster stride rate?

These questions are difficult to answer because of the number of variables involved, so you have to experiment. One clue to answering these questions may come from studying muscles.

As a muscle’s velocity of contraction increases, its maximum force of contraction decreases. So a muscle contracting slowly can produce greater force than the same muscle contracting at a faster rate. If the muscle’s velocity of contraction is multiplied by its maximum force of contraction for that velocity, the muscle’s power output for each velocity can be determined. The maximum power output occurs at a velocity approximately one-half the muscle’s maximum contraction velocity. The best choice of bicycling gear, stride length, and so on may be the one in which your muscles contract at a velocity corresponding to their velocity of maximal power output.

Another reason that power is an important topic in the study of human movement is that it is actually a constraint on human movement. Figure 4.5 shows the theoretical relationship between maximum power output and the duration of that power output for humans. This relationship shows the mechanical constraint placed on humans by their power-generating system (their metabolic system).

Summary

• Work done by a force is the force times the displacement of the object along the line of action of the force acting on it.
• Energy is defined as the capacity to do work.
• Energy takes two forms: potential energy and kinetic energy.
• The work done by a force on an object causes a change in energy of the object.
• Power is defined as the rate of doing work.