# Question Video: Finding an Unknown Component of the Displacement of a Particle given the Work Done by a Force Acting on It Mathematics

A force 𝐅 = (𝑚𝐢 − 9𝐣) N acts on a particle, causing a displacement 𝐬 = [−5𝐢 + (𝑚 + 6)𝐣] cm. If the work done by the force is 0.02 J, what is the value of 𝑚?

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### Video Transcript

A force 𝐅 equal to 𝑚𝐢 minus nine 𝐣 newtons acts on a particle, causing a displacement 𝐬 equal to negative five 𝐢 plus 𝑚 plus six 𝐣 centimetres. If the work done by the force is 0.02 joules, what is the value of 𝑚?

When the force and displacement are given in vector form as in this case, we can calculate the work done by working out the dot product of 𝐅 and 𝐬. The work done will be measured in joules, the force needs to be measured in newtons, and the displacement in metres. The force factor is in the correct newtons, so 𝐅 is equal to 𝑚𝐢 minus nine 𝐣. Our displacement is given in centimetres. To convert this to metres, we need to divide each component by 100 as there are 100 centimetres in one metre. The displacement vector in metres is therefore equal to negative five over 100 𝐢 plus 𝑚 plus six over 100 𝐣. This can be rewritten in decimal form as negative 0.05𝐢 plus 0.01𝑚 plus 0.06 𝐣.

We can now calculate the work done by finding the dot products of these two vectors. The dot product is found by firstly multiplying the coefficients of 𝐢. 𝑚 multiplied by negative 0.05 is equal to negative 0.05𝑚. We then multiply the coefficients of 𝐣. Negative nine multiplied by 0.01𝑚 is negative 0.09𝑚. And negative nine multiplied by 0.06 is negative 0.54. We were told in the question that the work done is 0.02. Therefore, this expression equals 0.02. Collecting like terms gives us negative 0.14𝑚 minus 0.54 is equal to 0.02. At this stage, we might notice that multiplying both sides by 100 will make the equation easier to deal with.

The equation simplifies to two is equal to negative 14𝑚 minus 54. We can then add 54 to both sides so that negative 14𝑚 is equal to 56. Finally, dividing both sides by negative 14 will give us our value of 𝑚. 56 divided by 14 is equal to four. Therefore, 56 divided by negative 14 is equal to negative four. The value of 𝑚 is negative four. We could substitute this back in to our initial expressions. The force is equal to negative four 𝐢 minus nine 𝐣. The displacement is equal to negative five 𝐢 plus two 𝐣 centimetres or negative 0.05𝐢 plus 0.02𝐣 metres.