Fundamental Principles

Conservative Force: The total work done in a closed loop is zero. $$ \oint \vec{F} \cdot d\vec{r} = W_{AB} + W_{BC} + W_{CA} = 0 $$

Non-Conservative Force: The total work done in a closed loop is not necessarily zero (it depends on the path). $$ W_{total} = W_{AB} + W_{BC} + W_{CA} \neq 0 \quad (\text{generally}) $$

(a) Conservative Force; $W_{AB} = W_{BC} \neq 0$

Using the loop rule:

$$ W_{AB} + W_{BC} + W_{CA} = 0 $$

Let $W_{AB} = W_{BC} = W$. Then:

$$ W + W + W_{CA} = 0 \implies W_{CA} = -2W $$

• (p) Is $W_{CA} = 0$? No, because $W \neq 0$.
• (q) Is $W_{CA} > 0$? Yes, if $W$ is negative, $W_{CA}$ is positive.
• (r) Is $W_{CA} = W_{BC}$? $-2W = W \implies W=0$, which contradicts the given.
• (s) Is $|W_{CA}| > |W_{BC}|$? $|-2W| = 2|W|$. Since $|W| > 0$, $2|W| > |W|$ is always true.

Matches: (q) [and strictly speaking (s) physically]

(b) Non-Conservative Force; $W_{AB} = W_{BC} \neq 0$

For non-conservative forces, the loop sum is arbitrary ($k$).

$$ W_{AB} + W_{BC} + W_{CA} = k $$

Let $W_{AB} = W_{BC} = W$. Then:

$$ 2W + W_{CA} = k \implies W_{CA} = k – 2W $$

Since $k$ (work done in the cycle) is not constrained to be zero, $W_{CA}$ can take various values.

• (p) Is $W_{CA} = 0$? Yes, if the cycle work $k = 2W$.
• (q) Is $W_{CA} > 0$? Yes, depending on $k$ and $W$.
• (r) Is $W_{CA} = W_{BC}$? Yes, if $k – 2W = W \implies k = 3W$.
• (s) Is $|W_{CA}| > |W_{BC}|$? Yes, possible for many values of $k$.

Matches: (p), (q), (r), (s)

(c) Conservative Force; $|W_{AB}| > |W_{BC}|$

From the loop rule:

$$ W_{CA} = -(W_{AB} + W_{BC}) $$

• (p) Is $W_{CA} = 0$? No, implies $W_{AB} = -W_{BC} \implies |W_{AB}| = |W_{BC}|$, which contradicts condition.
• (q) Is $W_{CA} > 0$? Yes, possible.
• (r) Is $W_{CA} = W_{BC}$? Possible. (e.g., $W_{AB} = -2W_{BC} \implies W_{CA} = -(-2W_{BC} + W_{BC}) = W_{BC}$. Condition $|-2W_{BC}| > |W_{BC}|$ holds).
• (s) Is $|W_{CA}| > |W_{BC}|$? Yes, possible. (e.g., same example as above).

Matches: (q), (r), (s)

(d) Non-Conservative Force; $|W_{AB}| > |W_{BC}|$

Again, the loop sum is arbitrary ($k$).

$$ W_{AB} + W_{BC} + W_{CA} = k $$ $$ W_{CA} = k – (W_{AB} + W_{BC}) $$

Due to the free parameter $k$ (energy loss/gain in the cycle), $W_{CA}$ is not tightly constrained by $W_{AB}$ and $W_{BC}$.

• (p) Is $W_{CA} = 0$? Yes, if $k = W_{AB} + W_{BC}$.
• (q) Is $W_{CA} > 0$? Yes.
• (r) Is $W_{CA} = W_{BC}$? Yes, if $k = W_{AB} + 2W_{BC}$.
• (s) Is $|W_{CA}| > |W_{BC}|$? Yes.

Matches: (p), (q), (r), (s)