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Answer :
- Calculate the force vector: $F = 20(0.6 i + 0.8 j) = 12 i + 16 j$.
- Calculate the displacement vector: $S = 4(2 i + 1 j) = 8 i + 4 j$.
- Calculate the work done: $W = F \cdot S = (12)(8) + (16)(4) = 96 + 64 = 160$.
- The work done is $\boxed{160 J}$.
### Explanation
1. Problem Analysis
We are given a force vector $F=20 N(0.6 i +0.8 j )$ and a displacement vector $S=4 m(2 i +1)$. We want to find the work done by the force in moving an object through the given displacement.
2. Work Done Formula
The work done $W$ by a force $F$ over a displacement $S$ is given by the dot product of the force and displacement vectors: $W = F "." S$.
3. Calculating Force Vector
First, let's find the force vector $F$:
$$F = 20(0.6 i + 0.8 j) = (20 \times 0.6) i + (20 \times 0.8) j = 12 i + 16 j$$
4. Calculating Displacement Vector
Next, let's find the displacement vector $S$:
$$S = 4(2 i + 1 j) = (4 \times 2) i + (4 \times 1) j = 8 i + 4 j$$
5. Calculating Work Done
Now, we can calculate the work done $W$:
$$W = F "." S = (12 i + 16 j) "." (8 i + 4 j) = (12 \times 8) + (16 \times 4) = 96 + 64 = 160$$
6. Final Answer
Therefore, the work done by the force $F$ in moving the object through the displacement $S$ is 160 J.
### Examples
In physics, calculating work done is essential for understanding energy transfer. For example, when pushing a box across a floor, knowing the force applied and the distance the box moves allows us to calculate the work done, which directly relates to the energy required for the movement. This concept is also crucial in engineering, such as designing engines or analyzing the efficiency of machines.
- Calculate the displacement vector: $S = 4(2 i + 1 j) = 8 i + 4 j$.
- Calculate the work done: $W = F \cdot S = (12)(8) + (16)(4) = 96 + 64 = 160$.
- The work done is $\boxed{160 J}$.
### Explanation
1. Problem Analysis
We are given a force vector $F=20 N(0.6 i +0.8 j )$ and a displacement vector $S=4 m(2 i +1)$. We want to find the work done by the force in moving an object through the given displacement.
2. Work Done Formula
The work done $W$ by a force $F$ over a displacement $S$ is given by the dot product of the force and displacement vectors: $W = F "." S$.
3. Calculating Force Vector
First, let's find the force vector $F$:
$$F = 20(0.6 i + 0.8 j) = (20 \times 0.6) i + (20 \times 0.8) j = 12 i + 16 j$$
4. Calculating Displacement Vector
Next, let's find the displacement vector $S$:
$$S = 4(2 i + 1 j) = (4 \times 2) i + (4 \times 1) j = 8 i + 4 j$$
5. Calculating Work Done
Now, we can calculate the work done $W$:
$$W = F "." S = (12 i + 16 j) "." (8 i + 4 j) = (12 \times 8) + (16 \times 4) = 96 + 64 = 160$$
6. Final Answer
Therefore, the work done by the force $F$ in moving the object through the displacement $S$ is 160 J.
### Examples
In physics, calculating work done is essential for understanding energy transfer. For example, when pushing a box across a floor, knowing the force applied and the distance the box moves allows us to calculate the work done, which directly relates to the energy required for the movement. This concept is also crucial in engineering, such as designing engines or analyzing the efficiency of machines.
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