A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by $\text F$ = $-\hat{i}+2\hat j+3\hat k$ where $\hat i,\hat j,\hat k$ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?

Question

A curve is given between the potential energy $U$ of a particle and its position on the $x$-axis as shown. Given: \[ \tan\theta_1 = 1,\qquad \tan\theta_2 = 3,\qquad \tan\theta_3 = -\frac{1}{2} \] If $F_{AB}$ is the force acting on the particle during motion from $A$ to $B$, similarly $F_{BC}$, $F_{CD}$ and $F_{DE}$ are the forces during $B$ to $C$, $C$ to $D$ and $D$ to $E$ respectively, arrange the magnitudes of these forces in decreasing order.

Answer 1

Given Data

Force: $\vec{F} = -\hat{i} + 2\hat{j} + 3\hat{k}$ (N, assumed)
Displacement: along z-axis, $\Delta z = 4$ m
Motion constrained: $\Delta x = \Delta y = 0$

$\vec{F} = -1\hat{i} + 2\hat{j} + \textbf{3}\hat{k}$

$\Delta \vec{r} = 0\hat{i} + 0\hat{j} + \textbf{4}\hat{k}$

Work Definition

Work = dot product of force and displacement:

$$W = \vec{F} \cdot \Delta \vec{r} = F_x \Delta x + F_y \Delta y + F_z \Delta z$$

Calculation

$$W = (-1)(0) + (2)(0) + (3)(4)$$ $$W = 0 + 0 + 12 = 12 \, \text{J}$$

Work Done

$W = \textbf{12 J}$

Vector Dot Product Verification

$$\vec{F} \cdot \Delta \vec{r} = |\vec{F}| |\Delta \vec{r}| \cos\theta$$ $$F_z = 3, \quad \Delta r_z = 4, \quad \cos 0^\circ = 1$$ $$W = 3 \times 4 \times 1 = 12 \, \text{J} \quad \checkmark$$

Key Physical Insight

Only z-component of force does work
x, y components perpendicular to motion → zero work
Constraint forces (preventing x,y motion) do no work
Result independent of path (straight z-line)

Component Table

Component	F (N)	Δr (m)	Work Contribution
x	-1	0	0 J
y	+2	0	0 J
z	+3	4	12 J

Solution and Explanation