Work, Energy, and Simple Machines · Lesson 13 of 13
Chapter Summary and Practice
“Work, energy and machines gather for one final meeting, and even the pulley has homework.”
• Connect work, energy, power and simple machines in one framework. • Recall and select the important formulas from the chapter. • Solve multi-step problems involving work and energy. • Interpret force, speed and potential-energy graphs. • Design investigations and simple-machine models using familiar materials.
Work provides a way to measure energy transferred by a force, energy describes the capacity to produce change, and power tells how rapidly the transfer occurs. Kinetic and potential energy let us analyse motion without following every force through every instant. Simple machines then show how the same work can be arranged through different combinations of force, direction and distance.
Detailed Chapter Summary
A constant force does work on an object when the object has displacement in the direction of that force. When force and displacement agree in direction, work is positive. When they oppose each other, work is negative. Work is zero when displacement is zero, force is zero, or force is perpendicular to displacement. A complete statement identifies the force doing the work and the object receiving it.
The joule is the SI unit of both work and energy. One joule is one newton metre. On a force-displacement graph, the area under the graph gives the work. A rectangle describes a constant force, while changing forces may require the area to be divided into simpler shapes.
The work-energy theorem connects a force interaction with an energy change. Positive net work increases the relevant energy of an object or system, while negative net work decreases it. Work is one way to transfer energy; heating, radiation, electric circuits and waves can also transfer energy.
Kinetic energy depends on mass and the square of speed. Doubling mass doubles kinetic energy at fixed speed, but doubling speed makes kinetic energy four times as large. This makes speed especially important in stopping and collision situations.
Potential energy can be stored by deforming an object or by changing the relative positions of interacting objects. Near Earth’s surface, raising an object through vertical height h increases the gravitational potential energy of the Earth-object system by mgh. The change depends on the initial and final heights, not on the path used.
Mechanical energy is the sum of kinetic and potential energy. During ideal free fall or ideal pendulum motion, one form decreases while the other increases by the same amount. The sum remains constant when no external resistive force changes the system’s mechanical energy. Friction and air resistance transfer part of it to thermal energy and sound.
Power distinguishes equal amounts of work completed in different times. More work in the same time or the same work in less time means greater power.
Simple machines change the magnitude or direction of effort. Mechanical advantage compares load with effort. A fixed pulley ideally has mechanical advantage one and mainly changes direction. An inclined plane reduces effort by increasing path length. A lever uses different arm lengths around a fulcrum to trade force for distance.
| Situation | First Question to Ask | Useful Relation |
|---|---|---|
| Force moves an object | What are the relative directions? | W = F × s with the correct sign |
| Speed changes | What are the initial and final kinetic energies? | W = ΔK |
| Height changes | What is the vertical rise or fall? | U = mgh |
| Gravity converts energy | Is friction negligible? | Kinitial + Uinitial = Kfinal + Ufinal |
| A task has a time interval | How rapidly is energy transferred? | P = W/t |
| A machine changes effort | What are load, effort and distances? | Mechanical advantage and work balance |
Bridging Science and Society
In hilly regions, a traditional watermill uses water flowing downhill to grind grain. Water begins at a greater height with gravitational potential energy. As it descends through a channel or pipe, potential energy becomes kinetic energy. The moving water turns a wheel, giving it rotational mechanical energy. A shaft carries this rotation to the grinding stone, which does useful work on grain.
A dam-based generating system follows the same central chain on a larger scale. Stored water descends, moving water turns turbines, and generators transform rotational mechanical energy into electrical energy. The technology differs in scale and purpose, but the work-energy reasoning remains connected.
At a Glance
• Work is done when a force produces displacement in relation to its direction. • Work may be positive, negative or zero. • Net work equals the change in energy. • Kinetic energy depends on mass and the square of speed. • Potential energy can arise from deformation or relative position. • Mechanical energy is conserved in ideal motion without external resistive work. • Power is work done per unit time. • Simple machines change force or direction but do not create energy. • Mechanical advantage compares load with effort.
Revise, Reflect, Refine
Problem
A 2 kg ball is thrown upward at 20 m s⁻¹ and reaches a maximum height of 19.4 m. Find the work done by air resistance using g = 10 m s⁻².
- 1.Initial kinetic energy = one-half × 2 × 20² = 400 J.
- 2.Choose initial potential energy as zero, so initial mechanical energy = 400 J.
- 3.At maximum height, speed is zero and kinetic energy is zero.
- 4.Final potential energy = mgh = 2 × 10 × 19.4 = 388 J.
- 5.Work by air resistance = final mechanical energy - initial mechanical energy.
- 6.Work by air resistance = 388 J - 400 J = -12 J.
- 7.The negative value shows that air resistance transferred 12 J away from mechanical energy.
Problem
A 1.5 kg coconut falls from a height of 10 m. Find its speed before impact and the depression made if sand provides an average resistive force of 3000 N. Use g = 10 m s⁻².
- 1.Initial potential energy = mgh = 1.5 × 10 × 10 = 150 J.
- 2.Neglecting air resistance, this becomes kinetic energy just before impact.
- 3.Set one-half mv² = 150 J.
- 4.One-half × 1.5 × v² = 150, so v² = 200.
- 5.Therefore, v = √200 ≈ 14.14 m s⁻¹.
- 6.During penetration, the coconut’s 150 J is used in work against sand.
- 7.Use work magnitude = force × depth: 150 = 3000 × d.
- 8.d = 150 ÷ 3000 = 0.05 m, or 5 cm.
Problem
A 60 kg rider accelerates a 100 kg scooter to speed v. The next day a 40 kg passenger joins, and the same speed is reached in the same time. Compare the ideal fuel energy used on the two days.
- 1.First-day moving mass = 60 + 100 = 160 kg.
- 2.Second-day moving mass = 60 + 40 + 100 = 200 kg.
- 3.At the same speed, kinetic energy is directly proportional to total mass.
- 4.Energy ratio = 160 : 200.
- 5.Simplify by dividing by 40: energy ratio = 4 : 5.
- 6.With the stated ideal assumption, fuel used is also in the ratio 4 : 5.
Problem
A force rises uniformly from 0 N to 50 N over 1 m, remains 50 N for 2 m, and then falls uniformly to 0 N over the final 1 m. Find the total work.
- 1.Divide the area under the graph into a left triangle, a rectangle and a right triangle.
- 2.Left triangle area = one-half × 1 m × 50 N = 25 J.
- 3.Rectangle area = 2 m × 50 N = 100 J.
- 4.Right triangle area = one-half × 1 m × 50 N = 25 J.
- 5.Total work = 25 + 100 + 25 = 150 J.
Problem
The same flag is raised to the same height first in 12 s and then in 6 s. Compare work and power.
- 1.The load and vertical height are unchanged, so the gravitational potential-energy gain is unchanged.
- 2.Therefore, the work required is the same in both cases.
- 3.Power equals work divided by time.
- 4.The second time is half the first time.
- 5.For equal work, halving time doubles power.
- 6.The second raising requires twice the average power.
Quiz
Which expression represents Work by a constant force along displacement?
Which expression represents Work-energy theorem?
Which statement correctly applies to the lesson “Chapter Summary and Practice”?
Which additional statement also correctly applies to the lesson “Chapter Summary and Practice”?
Which further statement also correctly applies to the lesson “Chapter Summary and Practice”?
Practice Problems
- State whether work is positive, negative or zero when gravity acts on a ball moving upward, moving downward and moving horizontally.
- Identify the energy transformations in a truck moving uphill, an unwinding watch spring, photosynthesis, water leaving a dam, a burning matchstick, a firecracker, a microphone, a glowing bulb and a solar panel.
- A crane raises a mass to twice the height in twice the time. Compare energy and power with the first lift.
- Draw a balanced seesaw for an adult twice as heavy as a child and label their distances from the fulcrum.
- A 10 kg block has 180 J of kinetic energy. Find its speed before any additional force acts.
- Explain how to use the area under a variable force-displacement graph to find a block’s final kinetic energy.
- An astronaut throws a ball upward with the same initial speed on Earth and the Moon, where gravity is one-sixth as strong. Compare the maximum heights.
- A car moving at constant speed is brought to rest by brakes. State the work done by the brakes and identify where the kinetic energy goes.
- A potential-energy graph is given for a ball on a frictionless track. Explain how total energy can be used to determine which positions are reachable and the speed at each reachable position.
- Explain why later peaks of a real roller-coaster model are lower than earlier peaks.
The Journey Beyond
Remove both ends of an empty pen barrel so its refill can slide freely. Attach a rubber band between the barrel and refill using a safe arrangement prepared with adult help. Stretch the band by different amounts and release it away from people and breakable objects. Compare the distance travelled by the refill. The model demonstrates elastic potential energy changing into kinetic energy.
Use only a light refill, point it toward an empty floor area, wear eye protection and never aim the device at a person, animal or fragile object.
Build a lever, pulley, inclined plane or a combination using rulers, cardboard, thread, cups and small weights. Measure the load and effort, calculate mechanical advantage and compare the movement distances. Record where friction or bending makes the real machine differ from the ideal prediction.
Interactive models can also help visualise energy that cannot be seen directly. Change mass, height, spring compression and friction in suitable simulations. Predict the outcome before each change, then compare energy bars, speed and position with the prediction.
The Quest Continues ...
Observations show that the Universe is expanding and that this expansion is accelerating. Scientists use the term dark energy for a proposed explanation of this large-scale behaviour. It is not a form that can be exchanged with the mechanical-energy systems studied here. Its effects are investigated through astronomical observations, and its nature remains an open question.
Work, energy and machines form one connected account: forces transfer energy through displacement, energy changes form, power measures the rate of transfer, and machines rearrange force and distance without creating energy.
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