Sketch the curve whose equation is \[y^2(1+x^2) = x^2(1-x^2),\] and find the area of a loop of the curve.
According to the Special Theory of Relativity, the dynamics of a particle, moving on a straight line, may be treated in a given frame of reference by solving the equation \[\frac{dp}{dt} = F\] where \(p\) is the momentum and \(F\) the force, the only difference between relativistic and ordinary mechanics being that the formula for the momentum is \[p = \frac{mv}{(1-v^2/c^2)^{1/2}}\] instead of \(mv\). Here \(m\) is the mass (a given constant), \(v\) is the velocity observed in that frame, and \(c\) the speed of light. Show that \[Fv = \frac{d}{dt}\left[\frac{mc^2}{(1-v^2/c^2)^{1/2}}\right].\] In the case where \(F\) is a constant force, and the particle starts from rest at the origin at time \(t = 0\), show that the distance covered after time \(t\) is \[x = \frac{c^2}{a}\left[\left(1 + \frac{a^2t^2}{c^4}\right)^{1/2} - 1\right],\] where \(a = F/m\). Give approximations to this result for \(at \ll c\) and \(at \gg c\) respectively, and comment on them.
Two astronomical bodies may be regarded as particles of masses \(M_1\) and \(M_2\), and attract each other according to the inverse square law. Prove that a possible solution of their equations of motion is one in which they move steadily on circles centred on their mass-centre, and give the relation between the radii and the period of rotation. Explain qualitatively why there are two tides per day rather than one.
An electric hand drill consists of a rigid casing held by the user, and in it are two parallel spindles \(S_1\) and \(S_2\) mounted on frictionless bearings. \(S_1\) is driven by a motor mounted rigidly in the casing. \(S_2\) incorporates the drill bit. The two spindles are coupled by gear wheels with \(n_1\) and \(n_2\) teeth respectively, which mesh externally without friction. The rotating parts are rigid, with moments of inertia \(I_1\) and \(I_2\) respectively about their axes, and their mass centres lying on the axes. The motor is designed to provide a constant torque with moment \(G\) (independent of its angular velocity). When drilling into a certain material, the bit experiences a resistive couple of moment \(k\omega_2\) where \(k\) is constant and \(\omega_2\) is the angular velocity of \(S_2\). At time \(t = 0\) the spindles are at rest and the motor is switched on to start drilling into that material. Show that at any later time \(t\) \[\omega _2 = \frac{n_2G}{n_1k}\left[1 - e^{-kt/A}\right],\] where \(A = I_2 + (n_2/n_1)^2I_1\). Find also, as a function of time, the couple which the user must exert to hold the casing stationary.
A uniform triangular lamina has mass \(M\) and sides \(a\), \(b\) and \(c\). Find its moment of inertia about the axis through its mass centre and perpendicular to its plane. [Hint: divide the triangle into four equal similar ones.]
A particle is projected in a given vertical plane from an origin \(O\), with velocity \((2gh)^{1/2}\). It passes through the point \((x, y)\) at time \(t\) after projection, the axes being horizontal and vertically upwards. Show that \[p^2 - 2(2h - y)p + r^2 = 0,\] where \(p = \frac{1}{2}gt^2\) and \(r^2 = x^2 + y^2\). Show that the points of the plane which are accessible from \(O\) by projection with the given velocity lie on or under the parabola having \(O\) as focus, and its vertex a distance \(h\) vertically above \(O\), and that the time taken to reach a point on this parabola is \((2v/g)^{1/2}\).
In three-dimensional Euclidean space, \(\mathbf{u}\) is a fixed vector of unit length, and \(\mathbf{r}\) is a given vector. Using the notation of scalar and vector products, show how to write the sum of a part parallel to \(\mathbf{u}\) and a part perpendicular to \(\mathbf{u}\). Hence, or otherwise, show that if the plane containing \(\mathbf{r}\) and \(\mathbf{u}\) is rotated through an angle \(\phi\) measured in the clockwise sense relative to the direction of \(\mathbf{u}\), and \(\mathbf{r}\) is thereby transported to a new position \(\mathbf{r}'\), then \[\mathbf{r}' = \mathbf{r}\cos\phi + \mathbf{u}(\mathbf{r} \cdot \mathbf{u})(1 - \cos\phi) + (\mathbf{u} \times \mathbf{r})\sin\phi.\]
A spacecraft may be regarded as a solid body which is convex (i.e. no straight line meets its surface more than twice), and its total surface area is \(A\). It is required to measure a certain type of radiation. If the radiation has a certain strength, and is unidirectional (i.e. incident on the spacecraft in the form of parallel rays), the response of the detector on board is given by \(Sk\), where \(S\) is the cross-sectional area presented by the spacecraft in that direction and \(k\) is a constant. If now the spacecraft is subjected to radiation of the same total strength but isotropic (i.e. scattered equally in all directions), show that the response is \(\frac{1}{4}Ak\), whatever the shape of the spacecraft.