15.2 Mechanics and fluids

A body moving through a medium experiences a retarding force that grows with its speed. For a force proportional to the speed, \(f = kV\), a body of mass \(m\) moving down a slope of inclination \(\theta\) obeys

\[m\frac{dV}{dt} = mg\sin\theta - kV,\qquad V(t) = \frac{mg\sin\theta}{k}\left(1 - e^{-kt/m}\right).\]

The speed rises monotonically towards the terminal value \(V_T = mg\sin\theta/k\), at which the driving force and the drag balance. Hence a body that has travelled further down the same slope is moving faster, and experiences a larger drag.

A rigid body rotating about an axis has a single angular speed \(\omega\), and a point at distance \(r\) from the axis moves with speed \(v = \omega r\). A cart turning on a circular track is such a body: its outer wheels travel a larger circle in the same time and must therefore turn faster than the inner ones. A wheel rolling without slipping covers a distance equal to the arc of its rim that touches the ground; one revolution of a wheel of diameter \(d\) moves it forward by \(\pi d\). Gears in mesh pass equal numbers of teeth across the point of contact, so two meshing wheels with \(n_1\) and \(n_2\) teeth return to their initial relative position after \(\operatorname{lcm}(n_1,n_2)\) teeth have passed.

The centre of mass of a system of parts is the mass-weighted mean of their positions. When the composition of a system changes, as when water drains from a bucket, the centre of mass need not move monotonically: it follows whichever part dominates the mass at each stage.

A floating body displaces its own weight of liquid (Archimedes’ principle). If ice of density \(\rho_i\) and volume \(V\) floats in water of density \(\rho_w\), the submerged volume is \(V_{\text{sub}} = \rho_i V/\rho_w\). When the ice melts, its mass \(\rho_i V\) becomes water of volume \(\rho_i V/\rho_w\), which is exactly \(V_{\text{sub}}\):

\[\boxed{\text{floating ice melts without changing the water level.}}\]

The lift on an aircraft wing is the upward aerodynamic force that supports its weight. It is upward throughout flight; what changes in climbing or descending is only its magnitude relative to the weight.

For a body executing simple harmonic motion \(\theta = \theta_0\sin\omega t\), the time spent in an interval \(d\theta\) is \(d\theta/\lvert\dot\theta\rvert\), and \(\lvert\dot\theta\rvert = \omega\sqrt{\theta_0^2 - \theta^2}\). The probability of finding the body in \(d\theta\) is therefore proportional to \(1/\sqrt{\theta_0^2 - \theta^2}\), which is least at the centre, where the body moves fastest, and greatest near the turning points, where it momentarily comes to rest.