A vector space is said to be finite-dimensional if there exists a finite number of vectors \(x_1, x_2, \ldots, x_n\) such that each vector in the space can be written as a linear combination $$c_1 x_1 + c_2 x_2 + \ldots + c_n x_n$$ with \(c_1, c_2, \ldots, c_n\) scalars. The vector space \(V_1\) consists of all sequences $$x = (\xi_1, \xi_2, \ldots)$$ of real numbers which have only a finite number of terms \(\xi_i\) non-zero. Addition and scalar multiplication are defined by $$(\xi_1, \xi_2, \ldots) + (\eta_1, \eta_2, \ldots) = (\xi_1 + \eta_1, \xi_2 + \eta_2, \ldots),$$ $$c(\xi_1, \xi_2, \ldots) = (c\xi_1, c\xi_2, \ldots).$$ \(V_2\) consists of all real sequences, the definitions of addition and scalar multiplication being the same. Prove that neither \(V_1\) nor \(V_2\) is finite-dimensional.
A particle of mass \(m\) moves in a plane under the action of a force of magnitude \(f(r)\) directed towards a fixed point \(O\) in the plane, where \(r\) is the distance of the particle from \(O\). Deduce from the equations of motion that the angular momentum about \(O\) and the total energy \(\frac{1}{2}mv^2 + \int f(r)dr\) remain constant. If \(f(r) = kr\), and initially \(r = r_0, v = v_0\) and the direction of motion is at right angles to the radius vector, find the value of \(r\) when the direction of motion is next at right angles to the radius vector.
A straight rigid uniform hair lies on a smooth table. At each end of the hair sits a flea. Show that, if the mass of the hair is not too great relative to that of the fleas, then in simultaneous jumps with the same velocity and angle of take-off they will be able to change ends without colliding in mid-air.
When an e.m.f. \(E(t)\) is applied to an inductor of constant inductance \(L\) and resistance \(R\), the current \(I\) is governed by the equation $$L \frac{dI}{dt} + RI = E.$$ Given that \(I = 0\) at time \(t_0\), find \(I(t)\) in the following cases: (i) \(E(t) = \begin{cases} 0 & \text{for } t < t_0 \\ E_0 \sin \omega t & \text{for } t > t_0 \end{cases}\) (\(E_0\) constant); (ii) \(E(t) = \begin{cases} 0 & \text{for } t < t_0 \\ E_0 \sin \omega t & \text{for } t_0 < t < t_1 \\ 0 & \text{for } t > t_1 \end{cases}\)
The measurement of a certain physical quantity \(Q\) involves the use of the unit of length. Let \(q\) denote the measure of \(Q\) when the unit of length is taken to be \(\mathbf{u}\), and \(\tilde{q}\) when it is taken to be \(\mathbf{\tilde{u}}\). Assume that, if \(\mathbf{u} = \lambda \mathbf{\tilde{u}}\), then $$\tilde{q} = f(\lambda, q).$$ Now suppose that the sum \(Q_1 + Q_2\) has a meaning independent of the choice of the unit of length. Then we must have $$q_1 + q_2 = q_3 \Rightarrow \tilde{q}_1 + \tilde{q}_2 = \tilde{q}_3.$$ Therefore $$f(\lambda, q_1) + f(\lambda, q_2) = f(\lambda, q_1 + q_2)$$ for all \(q_1, q_2\) and all positive \(\lambda\). Prove that \(f(\lambda, q)\) must be of the form \(\phi(\lambda)q\). By considering two successive changes of unit show that $$\phi(\lambda \lambda') = \phi(\lambda)\phi(\lambda')$$ and deduce the form of the function \(\phi(\lambda)\). (Assume that all the functions considered are differentiable.)
Three unequal rods \(A_0 A_1\), \(A_1 A_2\) and \(A_2 A_3\) are smoothly jointed at \(A_1\) and \(A_2\). The ends \(A_0\) and \(A_3\) can slide along a smooth horizontal rail. Find the position of stable equilibrium. Investigate the equilibrium of a similar system consisting of a chain of \(n\) rods \(A_0 A_1, \ldots, A_{n-1} A_n\), and show that there is precisely one stable configuration.
Assume that, if impulsive forces are applied to a rigid body at rest, the centre of mass \(G\) acquires a velocity \(F/M\) and the body acquires an angular velocity \(L/I\), where \(F\) is the resultant of the impulsive forces, \(L\) is their moment about \(G\), \(M\) is the mass, and \(I\) the moment of inertia about \(G\). Two uniform rods \(AB\), \(BC\) of masses \(M\), \(N\) and lengths \(2a\), \(2b\) respectively, smoothly jointed at \(B\), are placed in a straight line on a smooth horizontal surface. A horizontal impulsive force \(F\), at right angles to the line of the rods, is applied at \(A\). Find the initial speed \(V\) of \(C\). Show that if \(F\) is applied at \(C\) instead the initial speed of \(A\) is \(V\).
A particle is suspended from a fixed point by a light spring. If \(c\) is the extension of the spring when the particle hangs in equilibrium, and \(2\pi/\omega\) is the period of small vertical oscillations when equilibrium is disturbed, show that \(c\omega^2 = g\). From this particle is now suspended a second particle, of the same mass, by a similar spring. The particles are set in motion in a vertical line. Denoting the extensions of the upper and lower springs by \(2c + x\) and \(c + y\) respectively, write down the equations of motion. Show that two periodic motions each of the form $$x = a \cos \omega t, \quad y = b \cos \omega t$$ are possible, the frequencies being given by $$(\omega/\omega_0)^2 = \frac{1}{2}(3 \pm \sqrt{5}).$$ Find the corresponding values of \(b/a\).
A force \(\mathbf{F}\) acts at a point whose position vector from \(O\) is \(\mathbf{r}\). Define the moment of \(\mathbf{F}\) about \(O\) and the work done by \(\mathbf{F}\) in a displacement \(\delta \mathbf{r}\) of the point of application. A number of forces act at points of a rigid sheet of material, and are coplanar with the sheet. Deduce from your definitions the following. (If you express your definitions in terms of scalar or vector products, you should prove any properties of these products on which your deductions depend.) (i) If the sheet is given a uniform displacement \(\delta \mathbf{a}\) the total work done by the forces is equal to the work done by the resultant force \(\mathbf{F}\) in the displacement \(\delta \mathbf{a}\). (ii) If the sheet is given two uniform displacements \(\delta \mathbf{a}\) and \(\delta \mathbf{b}\) in succession, the total work done by the forces is equal to the work done by \(\mathbf{F}\) in the resultant displacement \(\delta \mathbf{a} + \delta \mathbf{b}\). (iii) If the sheet is given a small rotation \(\delta \theta\) about \(O\) the work done by the forces is \(L\delta \theta\), \(L\) being the total moment of the forces about \(O\). Find an expression for the work done by the forces in a small rotation \(\delta \theta\) about the point whose position vector is \(\mathbf{p}\).