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
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.