Work and Energy
Beginner
work-energy
Source: Principles of Physics
Problem Sets:
Work-energy-dynamics 1029
Problem
An ice block floating in a river is pushed through a displacement $\vec{d} = (15 \text{ m})\hat{i} - (12 \text{ m})\hat{j}$ along a straight embankment by rushing water, which exerts a force $\vec{F} = (210 \text{ N})\hat{i} - (150 \text{ N})\hat{j}$ on the block.
How much work does the force do on the block during the displacement?
W = 4950 J
The work done $W$ by a constant force $\vec{F}$ over a displacement $\vec{d}$ is given by the dot product of the two vectors. Given the force $\vec{F} = F_x\hat{i} + F_y\hat{j}$ and displacement $\vec{d} = d_x\hat{i} + d_y\hat{j}$, the work is calculated as:
$$W = \vec{F} \cdot \vec{d} = F_x d_x + F_y d_y$$Substituting the given values:
$$W = (210 \text{ N})(15 \text{ m}) + (-150 \text{ N})(-12 \text{ m})$$ $$W = 3150 \text{ J} + 1800 \text{ J}$$