What is the work done by the constant force F during the particle's displacement?

Steps to Solve

1. Identify the displacement and force vectors

The displacement vector is given by $$ \Delta r = 2.5 \hat{i} - 3.5 \hat{j} - 1.0 \hat{k} \quad (in , km) $$ and the force vector is $$ F = -1.0 \hat{i} + 2.0 \hat{j} + 1.0 \hat{k} \quad (in , N) $$.

2. Convert the displacement to meters

Since the displacement is given in kilometers, we need to convert it to meters: $$ \Delta r = 2.5 \times 1000 \hat{i} - 3.5 \times 1000 \hat{j} - 1.0 \times 1000 \hat{k} = 2500 \hat{i} - 3500 \hat{j} - 1000 \hat{k} \quad (in , m) $$.

3. Calculate the dot product of force and displacement

The work done by the force is calculated using the formula $$ W = F \cdot \Delta r $$ This can be expressed as $$ W = F_x \Delta r_x + F_y \Delta r_y + F_z \Delta r_z $$, which results in: $$ W = (-1.0)(2500) + (2.0)(-3500) + (1.0)(-1000) $$.

4. Perform the calculations

Now we can compute each term:

• The first term: $$ -1.0 \times 2500 = -2500 , J $$
• The second term: $$ 2.0 \times -3500 = -7000 , J $$
• The third term: $$ 1.0 \times -1000 = -1000 , J $$

Combine these to find the total work: $$ W = -2500 - 7000 - 1000 $$.

5. Find the final result

Calculating the total: $$ W = -2500 - 7000 - 1000 = -10500 , J $$.