Work, Energy and PowerIntroduction
Riya is late for the Rajdhani. She grabs her trolley bag by its long handle and pulls hard, but the bag barely speeds up. Her uncle takes the handle, lowers it until it is almost flat, and the same bag rolls off easily. Riya was not pulling any less. So where did her effort go?
In everyday talk we say a farmer ploughing a field, a porter carrying bricks and a student revising for an exam are all "working", and a marathon runner has a lot of "energy". In physics, work, energy and power have exact meanings, and those meanings only loosely match the everyday ones. This chapter builds all three, and the first tool we need is a way to ask how much of a force acts along the direction of motion.
Drag the end of the handle to change its angle with the floor. Riya always pulls with 50 N.
The blue arrow is Riya's whole pull, but only the amber arrow, the part along the floor, drives the bag forward. That part is F cos θ: all of it at θ = 0, half of it at θ = 60°, none of it at θ = 90°, where Riya only lifts. Multiplying one vector by the part of another along it is so useful that it has a name: the scalar product.
The Plan for This Chapter
We start with the scalar product of two vectors. With it we define the work done by a force and the kinetic energy of a moving body, and prove the work-energy theorem that links them, first for constant forces and then for forces that change. Next comes potential energy, energy stored by position, and the great rule that the total mechanical energy stays constant when only conservative forces act; springs are our main example. Finally, power measures how fast work is done, and collisions put momentum and energy to work together.
Riya pulls with 50 N at 60° to the floor. The part of her pull along the floor is
At 90° the force is at right angles to the motion, so it cannot speed the bag up or slow it down. Hold on to that idea: it is the key to when a force does no work.
The Scalar Product
You know from Chapter 3 that displacement, velocity, acceleration and force are vectors, and how to add and subtract them. There are two ways of multiplying two vectors. One, the vector product, gives a new vector and waits for Chapter 6. The other, the scalar product or dot product, gives a plain number. For two vectors A and B with an angle θ between them,
A, B and cos θ are all scalars, so A·B is a scalar too: it has a size and a sign, but no direction. Read it as "A dot B".
Drag the tips of A and B. The amber line is the part of B along A.
Writing B cos θ for the length of the amber line, the projection of B onto A, gives
So A·B is the size of the blue arrow A times the part of the green arrow B along it, or just as well B times the part of A along B. This picture is Fig. 5.1 of your book: (a) the two vectors and the angle between them, (b) the projection of B onto A, (c) the projection of A onto B.
The scalar product follows three familiar rules, where λ is any real number:
(commutative)
(distributive)
Dot Products of the Unit Vectors
The unit vectors î, ĵ and k̂ along the x, y and z axes each have length 1, and each is at 90° to the other two. So
In the same way and . Now write two vectors in components, and , and multiply out with the distributive law. Of the nine products, the six mixed ones are zero, which leaves
Two results follow straight away.
since a vector makes an angle of 0 with itself. And A·B = 0 whenever A and B are perpendicular.
Try it:
2 × 4 + 3 × (−1) = 8 − 3 = 5. Matching components multiply, the results add, and the unit vectors drop out. That is exactly what the dot lab computed in its last line.
Angle Between a Force and a Displacement
Example 5.1 A drone is pushed by the wind with a force N while its displacement is m. Find the angle between the force and the displacement, and the projection of the force on the displacement.
Solution: Using Eq. (5.1b),
=
The sizes come from Eq. (5.1c): F² = 16 + 4 + 16 = 36, so F =
so θ = cos⁻¹ 0.444 ≈
The projection of F on d is F cos θ = F·d / d = 8 / 3 ≈ 2.67 N. Of the 6 N push, only 2.67 N acts along the direction the drone actually moves.
Speed Round: Dot Products
Multiply matching components and add. The sign of A·B gives the angle away: positive means acute, zero means 90°, negative means obtuse.
Get five right to move on.