5.13. Points to Ponder

1. The phrase ‘calculate the work done’ is incomplete. We should refer (or imply clearly by context) to the work done by a specific force or a group of forces on a given body over a certain displacement.

When calculating work, it's important to specify which force is doing the work and over what distance. For example, if someone lifts a box, we need to clarify that we're talking about the gravitational force acting on the box while it is lifted a certain height. Clarifying these details ensures that the calculation is accurate and meaningful.

Imagine telling someone to calculate how much energy they used while doing housework. Without specifics, they might not realize they need to consider the force of their own muscles in lifting, the weight of the vacuum cleaner, or even how far they vacuumed.

2. Work done is a scalar quantity. It can be positive or negative unlike mass and kinetic energy which are positive scalar quantities. The work done by the friction or viscous force on a moving body is negative.

In physics, work is a scalar quantity, which means it has magnitude but no direction. Work can be positive when it enhances energy (like lifting a ball), and negative when it opposes energy (like friction slowing down a sliding object). This is significant because it can imply loss of energy in systems where work done is negative.

Think of pushing a heavy box across a floor. If the box is heavy and you push it, you are doing positive work. However, if there's friction, it might slow down despite your efforts, which can be thought of as negative work. It's akin to running against a strong wind – while you're exerting effort, the wind makes it harder for you to move forward.

3. For two bodies, the sum of the mutual forces exerted between them is zero from Newton’s Third Law, F12 + F21 = 0. But the sum of the work done by the two forces need not always cancel, i.e., W12 + W21 ≠ 0. However, it may sometimes be true.

According to Newton’s Third Law, if body A exerts a force on body B, body B exerts an equal and opposite force on body A. While this law dictates the balance of forces, the work done by each force on the other can differ significantly. This means one force could do positive work while the other does negative work, depending on the situation.

Imagine two ice skaters pushing off each other. They exert equal forces, but if one skater moves further than the other, the work done by the pushing force is not the same. One skater gets pushed back farther, implying that one has traveled more distance than the other, highlighting the imbalance in work done.

4. The work done by a force can be calculated sometimes even if the exact nature of the force is not known. This is clear from Example 5.2 where the WE theorem is used in such a situation.

In certain contexts, such as when the work-energy theorem is applied, one can calculate the work done even without knowing all details about the forces involved. The theorem states that the work done is equal to the change in kinetic energy. Therefore, we can make calculations based on the energies before and after an event, irrespective of intermediary forces. It simplifies situations where complex dynamics are present.

If you know how fast a car was going before a crash and how fast it was going after, you can still calculate the work done in that crash without needing all the specifics of the forces acting on it. It’s like calculating how much a hill increased your speed on a bike ride without knowing the exact incline.

5. The WE theorem is not independent of Newton’s Second Law. The WE theorem may be viewed as a scalar form of the Second Law. The principle of conservation of mechanical energy may be viewed as a consequence of the WE theorem for conservative forces.

The work-energy theorem links work done to changes in kinetic energy, which is essentially an extension of Newton's Second Law. The conservation of energy, which states that total mechanical energy remains constant when only conservative forces act, derives from this relationship because it implies that any work done transforms energy forms but doesn't create or destroy energy.

Think of a rollercoaster. As it climbs (doing work against gravity), it gains potential energy. When it comes down, that potential energy converts back into kinetic energy as it speeds up. Both principles of force application and energy conservation hand-in-hand describe the ride experience.

6. The WE theorem holds in all inertial frames. It can also be extended to non-inertial frames provided we include the pseudoforces in the calculation of the net force acting on the body under consideration.

The work-energy theorem applies universally in all inertial frames where laws of motion are straightforward. Even in accelerated (non-inertial) frames, we can still use it by accounting for fictitious forces, like considering how the feeling of being pushed back in your seat in a speeding car is just a result of acceleration.

Imagine you're in a moving elevator. You feel heavier when it accelerates upward. If you want to calculate how hard you’re pushing against the floor, it’s more complex because that acceleration needs to be considered as an additional force acting on you.

7. The potential energy of a body subjected to a conservative force is always undetermined up to a constant. For example, the point where the potential energy is zero is a matter of choice. For the gravitational potential energy mg h, the zero of the potential energy is chosen to be the ground. For the spring potential energy kx2/2, the zero of the potential energy is the equilibrium position of the oscillating mass.

Potential energy depends on reference points. When calculating gravitational or spring potential energy, we choose where this energy is considered 'zero'. This means that potential energy is relative – it can change based on the chosen reference point, allowing us flexibility based on the problem being addressed.

Think about a book on a shelf. If we say the shelf is two meters above the ground, then the potential energy is based on that height. If we move the shelf up to three meters and don't adjust our zero-point, the energy would increase without ever adding any additional height from the perspective of the book itself – it’s simply a product of where we define our reference point.

8. Every force encountered in mechanics does not have an associated potential energy. For example, work done by friction over a closed path is not zero and no potential energy can be associated with friction.

Not all forces allow for potential energy calculations. Non-conservative forces like friction convert mechanical energy to thermal energy, and thus do not allow for a simple potential energy description since there’s energy loss to heat. The energy lost in overcoming friction cannot be stored as potential energy.

Imagine sliding down a hill. If there's friction, you lose energy as heat – that’s energy that can’t be recovered like potential energy could be if you were at the top of the hill before descending. That loss from friction is akin to taking one step forward in your uphill battle only to slide back down again.

9. During a collision: (a) the total linear momentum is conserved at each instant of the collision; (b) the kinetic energy conservation (even if the collision is elastic) applies after the collision is over and does not hold at every instant of the collision. In fact, the two colliding objects are deformed and may be momentarily at rest with respect to each other.

In collisions, momentum is always conserved, which means total momentum before the collision equals total momentum afterward. However, kinetic energy may not be conserved, particularly in inelastic collisions where objects may deform and energy is dissipated. This means while they might briefly stop moving relative to one another, momentum remains unchanged throughout the interaction.

In a car crash, while the cars may crumple and energy is lost, the total momentum of both vehicles before and after the crash can still be accounted for. It’s like a dance where partners may lose sync but still keep the rhythm of the dance overall!

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