10273 problems found
Explain how to reduce the solution of a dynamical problem to that of a statical problem. A uniform rod of length \(l\) attached at one end to a fixed point by a smooth universal joint rotates freely under gravity as a conical pendulum. If \(\omega\) is the angular velocity of the vertical plane through the rod, and \(\alpha\) the constant inclination of the rod to the vertical, prove that \[ \omega^2 = \frac{3g}{2l}\sec\alpha. \]
Tangents \(q_1, q_2, q_3\), are drawn at three points \(P_1, P_2, P_3\) on the parabola \(y^2 = 4ax\), and \(Q_1, Q_2, Q_3\) are the vertices of the triangle formed by \(q_1, q_2, q_3\) (\(Q_1\) being opposite to \(q_1\), etc.). Through \(Q_1\) are drawn lines parallel to \(q_1\) and to \(P_2 P_3\); and similarly for the other vertices. Prove that the six lines thus obtained all touch the parabola \[ (y - 2as_1)^2 + 8a(x-as_2) = 0, \] where \[ s_1 = t_1+t_2+t_3, \quad s_2 = t_2 t_3 + t_3 t_1 + t_1 t_2, \] \(t_1, t_2, t_3\) being the parameters of \(P_1, P_2, P_3\) in the parametric representation \(x=at^2, y=2at\).
Find \[ \text{(i) } \int \frac{dx}{x^2\sqrt{x^2+1}}, \quad \text{(ii) } \int_0^\infty \frac{xdx}{(1+x^2)^2}; \] and shew that, if \(a\) and \(b\) are positive, \[ \int_0^\pi \frac{\sin^2x\,dx}{a^2 - 2ab \cos x + b^2} = \frac{\pi}{2a^2} \quad \text{or} \quad \frac{\pi}{2b^2}, \] according as \(a\) is greater or less than \(b\).
A particle of mass \(m\) resting on the highest point of a fixed sphere of radius \(a\) and coefficient of friction \(\frac{1}{4}\) is slightly disturbed and slides down the sphere in a vertical plane. Prove that when the radius to the particle makes an angle \(\theta\) with the vertical, the angular velocity, \(\dot{\theta}\), of the radius is given by the equation \[ \frac{d}{d\theta}(\dot{\theta}^2 e^{-\theta}) = \frac{g}{a}(2\sin\theta - \cos\theta)e^{-\theta}, \] and shew that the normal reaction on the sphere at this instant is \[ \frac{mg}{2}[3(\cos\theta+\sin\theta)-e^\theta]. \]
A particle of mass \(m\) is describing an orbit in a plane under a force \(\mu m r\) towards a fixed point at a distance \(r\). Taking this point as origin of coordinates, shew that, if when the particle is at a point \((a,b)\) it has a velocity with components \(u,v\) parallel to the axes, the orbit will be given by \[ \mu(bx-ay)^2 + (vx-uy)^2 = (av-bu)^2. \]
Interpret the equation \(S - \lambda uu' = 0\), where \(\lambda\) is a constant and \begin{align*} S &= ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \\ u &= lx+my+nz=0, \quad u' = l'x+m'y+n'z=0, \end{align*} are the equations, in a system of homogeneous coordinates in a plane, of a conic and two straight lines. In a plane are given a triangle \(ABC\), a conic \(S\) not passing through a vertex of \(ABC\), and a point \(O\) not on a side of \(ABC\). A variable line through \(O\) meets \(S\) in \(P\) and \(Q\), and the conic through \(A, B, C, P, Q\) cuts \(S\) again in \(X\) and \(Y\). Prove that \(XY\) touches a fixed conic \(\Gamma\). Shew also that, for given \(ABC\) and \(O\), two different conics \(S\) and \(S'\) will give the same conic \(\Gamma\) provided that a conic can be drawn through \(A, B, C\) and the four points of intersection of \(S\) and \(S'\).
Draw the curves
Each of six similar particles is of weight \(w\), and is attached to a point \(O\) by a light inextensible string of length \(l\). Any two particles repel each other with a force equal to \(\lambda\) times their distance apart (\(\lambda\) is greater than \(w/6l\)). Prove that when the particles are in equilibrium at the vertices of a regular hexagon, the length of a side of the hexagon is \[ (l^2 - w^2/36\lambda^2)^{\frac{1}{2}}. \]
A uniform rectangular block, whose edges are of length \(2a, 2b, 2c\), and whose weight is \(w\), rests in equilibrium on a rough plane inclined to the horizontal at an angle \(\alpha\). The edges of length \(2c\) are horizontal, and those of length \(2a\) are parallel to a line of greatest slope. If \(\mu\) is the coefficient of friction, shew that \(\mu > \tan\alpha\) and \(b\tan\alpha < a\). A string is attached to the middle of the lower horizontal edge of the face parallel to, but not in contact with, the plane. It is parallel to a line of greatest slope, and passes over the middle of the upper horizontal edge. The tension in the string is gradually increased from zero. Shew that the block slips or tilts when equilibrium is first broken according as \[ a \lesseqgtr b(2\mu + \tan\alpha). \]
State necessary and sufficient conditions for a system of forces in one plane to be in equilibrium. Forces whose magnitudes are proportional to the sides of a triangle act inwards perpendicular to the sides at their middle points. Prove that the system is in equilibrium. If the lines of action of the forces be all rotated in the same sense through the same angle, each about the point where it meets its side of the triangle, shew that the system is equivalent to a couple, and find its magnitude.