2019 NYJC P1 Q12

In the study of force field, we are often interested in whether the work done in moving an object from point \({A}\) to point \({B}\) is independent of the path taken. If a force field is such that the work done is independent of the path taken, it is said to be a conservative field.

A force field \({\mathbf{F}}\) can be regarded as a vector \({\mathbf{F}=M\left( x,y \right)\mathbf{i}+N\left( x,y \right)\mathbf{j}}\) where \(M\) and \(N\) are functions of \(x\) and \(y\). The path that the object is moving along is denoted by \({C}\). The work done in moving the object along the curve \({C}\) from the point where \({x=a}\) to the point where \(x=b\) is given by

\(W = \int_{a}^{b} \left[ M(x, y) + N(x, y) \frac{\mathrm{d}y}{\mathrm{d}x} \right] \mathrm{d}x,\)

where \(y=\mathrm{f}\left( x \right)\) is the equation of the curve \({C}\).

1. Sketch the curve \({C}\) with equation \({{{y}^{2}}=4\left( 1-x \right)}\), for \(x\le 1\).[2]
2. Find an expression of \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \({y}\).[1]

3. The points \({P}\) and \({Q}\) are on \({C}\) with \({x=1}\) and \({x=-3}\) respectively and \({Q}\) is below the \(x\)-axis.
   
   Find the equation of the line \(PQ\).

   [2]

For the rest of the question, the force field is given by \({\mathbf{F}={{x}^{2}}\mathbf{i}+x{{y}^{2}}\mathbf{j}}\).

4. Show that the work done in moving an object along the curve \({C}\) from \({Q}\) to \(P\) is given by the integral \(\int_{-3}^{1} \left( x^2 + 4x(1 - x)^{\frac{1}{2}} \right) \mathrm{d}x\). Hence evaluate the exact work done in moving the object along the curve \({C}\) from \({Q}\) to \(P\).

   [4]
5. Find the work done in moving an object along the line \({PQ}\) from \({Q}\) to \(P\) to \({2}\) decimal places.[2]
6. Determine, with reason, whether \({\mathbf{F}}\) is a conservative force field.[1]