Question Video: Finding the Change in a Body’s Potential Energy Over a Time Period | Nagwa Question Video: Finding the Change in a Body’s Potential Energy Over a Time Period | Nagwa

Question Video: Finding the Change in a Body’s Potential Energy Over a Time Period Mathematics

A body is moving under the action of a constant force 𝐅 = (5𝐢 + 3𝐣) N, where 𝐢 and 𝐣 are two perpendicular unit vectors. At time 𝑡 seconds, where 𝑡 ≥ 0, the body’s position vector relative to a fixed point is given by 𝐫 = [(𝑡² + 4)𝐢 + (4𝑡² + 8)𝐣] m. Determine the change in the body’s potential energy in the first 9 seconds.

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Video Transcript

A body is moving under the action of a constant force 𝐅, which is equal to five 𝐢 plus three 𝐣 newtons, where 𝐢 and 𝐣 are two perpendicular unit vectors. At time 𝑡 seconds, where 𝑡 is greater than or equal to zero, the body’s position vector relative to a fixed point is given by 𝐫 is equal to 𝑡 squared plus four 𝐢 plus four 𝑡 squared plus eight 𝐣 meters. Determine the change in the body’s potential energy in the first nine seconds.

Due to the conservation of energy and the work–energy principle, we know that the sum of the change in potential energy and the work done is equal to zero. This is because energy can only be transferred. It cannot be created or destroyed. In this case, we are trying to calculate the change in potential energy.

We know that work done is equal to force multiplied by displacement. And when dealing with vectors, we find the dot product of the force vector and displacement vector. We are told that the force is equal to five 𝐢 plus three 𝐣 newtons. At present, the displacement is unknown. We are given the position vector of the body. And we are interested in the change in potential energy in the first nine seconds. This means that we need to calculate the position vector when 𝑡 equals zero and 𝑡 equals nine.

When 𝑡 is equal to zero, we have zero squared plus four 𝐢 plus four multiplied by zero squared plus eight 𝐣. This simplifies to four 𝐢 plus eight 𝐣. When 𝑡 is equal to nine, the position vector is equal to nine squared plus four 𝐢 plus four multiplied by nine squared plus eight 𝐣. This is equal to 85𝐢 plus 332𝐣.

We can then calculate the displacement vector by subtracting the initial position from the final position. 85𝐢 minus four 𝐢 is equal to 81𝐢, and 332𝐣 minus eight 𝐣 is 324𝐣. The displacement of the body in the first nine seconds is 81𝐢 plus 324𝐣.

We can now calculate the dot product of the force and displacement. This is equal to the sum of five multiplied by 81 and three multiplied by 324. This is equal to 405 plus 972, which gives us a total work done of 1,377.

We can now use this value to calculate the change in potential energy. As this value is positive, we know the change in potential energy will be negative. The GPE plus 1,377 must equal zero. This means that the change in potential energy is negative 1,377 joules. The body’s potential energy has decreased by 1,377 joules in the first nine seconds.

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