Question Video: Finding the Work Done by a Force in Vector Form Acting on a Body Moving between Two Points | Nagwa Question Video: Finding the Work Done by a Force in Vector Form Acting on a Body Moving between Two Points | Nagwa

Question Video: Finding the Work Done by a Force in Vector Form Acting on a Body Moving between Two Points Mathematics • Third Year of Secondary School

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A body moves under the force 𝐅 = 6𝐒 βˆ’ 9𝐣 from point 𝐴(βˆ’2, 8) to point 𝐡(1, βˆ’7). Determine the work done by the force, where the displacement is measured in meters and the force in newtons.

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Video Transcript

A body moves under the force 𝐅 equal to six 𝐒 minus nine 𝐣 from the point 𝐴 negative two, eight to the point 𝐡 one, negative seven. Determine the work done by the force, where the displacement is measured in meters and the force in newtons.

We begin by recalling that the work done by any force is equal to the scalar or dot product of the force vector and the displacement vector 𝐝. In this question, we’re told that the force vector is equal to six 𝐒 minus nine 𝐣 and that this is measured in newtons. We are not given the displacement vector 𝐝. However, we are told the force moves the body from point 𝐴 to point 𝐡. The displacement vector is therefore equal to the vector 𝐀𝐁. And we can calculate this by subtracting the position vector of point 𝐴 from the position vector of point 𝐡.

We have 𝐒 minus seven 𝐣 minus negative two 𝐒 plus eight 𝐣, and this is equal to three 𝐒 minus 15𝐣. The displacement vector is equal to three 𝐒 minus 15𝐣 meters. We can now determine the work done by finding the scalar product of our two vectors. This is the scalar product of six 𝐒 minus nine 𝐣 and three 𝐒 minus 15𝐣. We calculate this by finding the sum of the products of the components. We have six multiplied by three plus negative nine multiplied by negative 15. This simplifies to 18 plus 135, which is equal to 153.

Since the force and the displacement are measured in standard units of newtons and meters, the work done is 153 joules.

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