The power utilised when a force of \((2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{N}\)…
The power utilised when a force of \((2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{N}\) acts on a body for \(4 \mathrm{~s}\), producing a displacement of \((3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) \mathrm{m}\), is
\(9.5 \mathrm{~W}\)
\(7.5 \mathrm{~W}\)
\(6.5 \mathrm{~W}\)
\(4.5 \mathrm{~W}\)
Solution
Force, \(\mathbf{F}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\)
Displacement, \(\mathbf{s}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\)
\(\begin{aligned}
& \therefore \text { Velocity } =\frac{\text { Displacement }}{\text { Time }} \\
& \Rightarrow v =\frac{\mathbf{s}}{t}=\frac{1}{4}(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})
\end{aligned}\)
\(\therefore\) Power utilised is given as
\(\begin{aligned}
P & =\mathbf{F} \cdot \mathbf{v} \\
& =(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \cdot \frac{1}{4}(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) \\
& =2 \times \frac{3}{4}+3 \times \frac{4}{4}+4 \times \frac{5}{4} \\
& =1.5+3+5=9.5 \mathrm{~W}
\end{aligned}\)