Work, Energy and Power: common doubts, answered
The questions students ask most often about Work, Energy and Power, each with a short answer. For the full chapter, read the Work, Energy and Power notes.
The scalar product
Read this section in the notes →What is the difference between the dot product and ordinary multiplication?
The dot product multiplies two vectors and gives a scalar, A·B = AB cos θ, so it depends on the angle between them. It equals the product of one vector's magnitude and the component of the other along it. When the vectors are perpendicular the dot product is zero, however large they are; ordinary multiplication of their sizes would not be.
How do you find the angle between two vectors using the dot product?
Use cos θ = A·B / (AB). Compute A·B from components as AxBx + AyBy + AzBz, find each magnitude from its components, and divide. If the dot product comes out zero the vectors are perpendicular; if negative, the angle between them is more than 90°.
Work
Read this section in the notes →Can work done be negative?
Yes. Work is W = Fd cos θ, so it is negative whenever the force has a component opposite to the displacement, meaning θ is more than 90°. Kinetic friction on a sliding block and gravity on a ball thrown upward both do negative work. Negative work takes kinetic energy away from the body.
Why is no work done when you carry a bag horizontally?
In physics, work needs a force component along the displacement. Holding a bag means pushing upward, while the bag moves horizontally, so the angle is 90° and cos 90° = 0. You feel tired because your muscles are continually contracting, but that is internal physiological effort, not mechanical work done on the bag.
Kinetic energy and the work-energy theorem
Read this section in the notes →What does the work-energy theorem say?
It says the net work done by all forces on a body equals the change in its kinetic energy: K_f − K_i = W_net. It follows from Newton's second law and holds for any forces, constant or variable, conservative or not. It is often the quickest route when a problem gives speeds and distances but no times.
If momentum increases by 30%, does kinetic energy also increase by 30%?
No. Since K = p²/2m, kinetic energy depends on the square of momentum. A rise of 30% in momentum multiplies it by 1.3, so kinetic energy is multiplied by 1.69, an increase of 69%. Small percentage changes roughly double, but for larger ones always square the factor rather than doubling the percentage.
Can kinetic energy be negative?
No. Kinetic energy is ½mv², and both mass and the square of the speed are never negative, so it is zero or positive. Potential energy, by contrast, can be negative because only differences in it matter and the zero level is chosen freely.
Work done by a variable force
Read this section in the notes →How do you find work done by a variable force?
Integrate the force over the displacement: W = ∫ F(x) dx from x_i to x_f. Graphically, this is the area under the force-displacement curve between the two positions. Areas below the axis count as negative work. You cannot use W = Fd unless the force is constant over the whole path.
Potential energy and conservative forces
Read this section in the notes →What is a conservative force?
A conservative force is one whose work depends only on the starting and ending points, not on the path, so its work around any closed loop is zero. Gravity and the spring force are conservative, and each has a potential energy defined for it. Friction is not conservative because a longer path means more energy lost.
Why is there no potential energy for friction?
Because friction is non-conservative: the work it does depends on the path and cannot be recovered by retracing it. Potential energy can be defined only when work depends on the end points alone, so that energy is stored and can be returned. Energy lost to friction becomes heat and is not stored as potential energy.
Conservation of mechanical energy
Read this section in the notes →When is mechanical energy conserved?
Total mechanical energy, kinetic plus potential, stays constant when only conservative forces do work. Then K_i + V_i = K_f + V_f. If friction or air drag does work, mechanical energy falls and the lost amount appears as heat, so using mgh = ½mv² on a rough path gives the wrong speed.
What is the minimum speed needed at the bottom to complete a vertical circle?
For a bob on a string of length L, the minimum speed at the bottom is √(5gL). At the top, tension can fall to zero but gravity alone must provide the centripetal force, so the speed there must be at least √(gL). Energy conservation between top and bottom, a height 2L apart, then gives √(5gL) at the bottom.
Potential energy of a spring
Read this section in the notes →Why is the potential energy of a spring ½kx² and not kx?
Because the spring force grows from zero as the spring is stretched, so the force is not constant. The work needed is the area under the force-extension graph, a triangle of base x and height kx, which gives ½kx². Using kx times x would double the true stored energy.
Is the work done by a spring positive or negative?
It depends on whether the spring is being stretched or allowed to relax. While you stretch or compress a spring, the spring force opposes the displacement and does negative work, W_s = −½k(x_f² − x_i²). When the spring returns towards its natural length, it does positive work on the attached body.
Power
Read this section in the notes →Is the kilowatt hour a unit of power?
No, the kilowatt hour is a unit of energy. It is the energy used by a device of power one kilowatt running for one hour, which is 3.6 × 10⁶ J. Electricity bills charge for energy consumed, which is why they use kWh, while the kilowatt itself measures power, the rate of using energy.
Collisions
Read this section in the notes →What is the difference between elastic and inelastic collisions?
In both, total momentum is conserved if no external force acts. In an elastic collision total kinetic energy is also the same before and after; in an inelastic one some kinetic energy turns into heat, sound or deformation. A perfectly inelastic collision is the extreme case where the bodies stick together and the loss of kinetic energy is greatest.
Is kinetic energy conserved during an elastic collision?
Only before and after the collision, not during it. While the bodies are in contact they deform, and part of the kinetic energy is stored briefly as elastic potential energy. Once they separate, that energy is fully returned. This is why elastic collisions are defined by comparing the initial and final kinetic energy alone.
What happens when a ball hits an identical ball at rest elastically?
In a head-on elastic collision between equal masses, the moving ball stops and the ball at rest moves off with the first ball's velocity. The formula v₁f = (m₁ − m₂)v₁ᵢ/(m₁ + m₂) gives zero when the masses are equal. The bodies exchange velocities, which is what you see with a line of identical swinging balls.
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