6 The physical quantity pressure, denoted by \(P\), can be calculated using the formula \(P = \frac { F } { A }\) where \(F\) is a force and \(A\) is an area.
- Find the dimensions of \(P\).
An object of mass \(m\) is moving on a smooth horizontal surface subject to a system of forces which begin to act at time \(t = 0\). The initial velocity of the object is \(u\) and its velocity and acceleration at time \(t\) are denoted by \(v\) and \(a\) respectively. The object exerts a pressure \(P\) on the surface. The total work done by the forces is denoted by \(W\).
A Mathematics class suggests three formulae to model the quantity \(W\).
The first suggested formula is \(W = \frac { 1 } { 2 } m v ^ { 2 } - \frac { 1 } { 2 } m u ^ { 2 } + m P\). - Use dimensional analysis to show that this formula cannot be correct.
The second suggested formula is \(W = k u ^ { \alpha } v ^ { \beta } t ^ { \gamma }\) where \(k\) is a dimensionless constant.
- Use dimensional analysis to show that this formula cannot be correct for any values of \(\alpha , \beta\) and \(\gamma\).
The third suggested formula is \(W = k u ^ { \alpha } a ^ { \beta } m ^ { \gamma } t ^ { \delta }\) where \(k\) is a dimensionless constant.
- Explain why it is not possible to use dimensional analysis to determine the values of \(\alpha\), \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
- Given that \(\alpha = 3\), use dimensional analysis to determine the values of \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
- By considering what the formula predicts for large values of \(t\), explain why the formula derived in part (d)(ii) is likely to be incorrect.