Question Video: Finding the Unknown Component of a Force Acting on a Body given the Change in Its Potential Energy | Nagwa Question Video: Finding the Unknown Component of a Force Acting on a Body given the Change in Its Potential Energy | Nagwa

# Question Video: Finding the Unknown Component of a Force Acting on a Body given the Change in Its Potential Energy Mathematics

A body is moving in a straight line from point ๐ด (โ6, 0) to point ๐ต (โ5, 4) under the action of the force ๐ = (๐๐ข + 2๐ฃ) N. Given that the change in the bodyโs potential energy is 2 joules and that the displacement is in meters, determine the value of the constant ๐.

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### Video Transcript

A body is moving in a straight line from point ๐ด negative six, zero to point ๐ต negative five, four under the action of the vector force ๐, which is equal to ๐๐ข plus two ๐ฃ newtons. Given that the change in the bodyโs potential energy is two joules and that the displacement is in meters, determine the value of the constant ๐.

We are told that the body moves in a straight line from point ๐ด to point ๐ต, where ๐ด and ๐ต have coordinates negative six, zero and negative five, four. This means that we move one unit to the right and four units up. If we consider the unit vectors ๐ข and ๐ฃ in the horizontal and vertical direction, respectively, our displacement vector is equal to ๐ข plus four ๐ฃ.

We are also told that the vector force acting on the body is ๐๐ข plus two ๐ฃ newtons. We know that the work done is the dot or scalar product of the force vector and the displacement vector. The work done is therefore equal to the dot product of ๐ข plus four ๐ฃ and ๐๐ข plus two ๐ฃ.

To calculate the dot product, we find the sum of the products of the individual components. In this question, this is equal to one multiplied by ๐ plus four multiplied by two. The ๐ข-components are one and ๐, and the ๐ฃ-components are four and two. This simplifies to ๐ plus eight.

We are also told in the question that the change in potential energy is equal to two joules. As energy can only be transferred and not destroyed or created, we know that the sum of the work done and the gravitational potential energy is equal to zero. This means that ๐ plus eight plus two must equal zero. Collecting like terms, we have ๐ plus 10 is equal to zero. Finally, we can subtract 10 from both sides of this equation, giving us a value of ๐ equal to negative 10. This means that the vector force ๐ is equal to negative 10๐ข plus two ๐ฃ.