10273 problems found
If \(u, v\) are positive and \(p>1\), shew that
\[\frac{u^p}{v^{p-1}} \ge pu - (p-1)v.\]
By writing first \(u = \frac{x}{x+y}, v = \frac{\xi}{\xi+\eta}\) and then \(u = \frac{y}{x+y}, v=\frac{\eta}{\xi+\eta}\) with positive \(x,y,\xi,\eta\) deduce that
\[ \frac{x^p}{\xi^{p-1}} + \frac{y^p}{\eta^{p-1}} \ge \frac{(x+y)^p}{(\xi+\eta)^{p-1}}. \]
Now substitute \(x=a_k, y=y_k, \xi = \sum_{k=1}^n a_k, \eta = \sum_{k=1}^n y_k\) and deduce
\[ \left(\sum_{k=1}^n a_k^p\right)^{\frac{1}{p}} + \left(\sum_{k=1}^n y_k^p\right)^{\frac{1}{p}} \ge \left\{\left(\sum_{k=1}^n(a_k+y_k)\right)^p\right\}^{\frac{1}{p}}.\]
Under what conditions does equality hold?
What is the corresponding result when \(0
Shew how to find \(\lim_{x\to 0} \frac{f(x)}{g(x)}\), when \(f(0)=0\) and \(g(0)=0\). Find the limit as \(x \to 0\) of
An elastic string of modulus \(\lambda\) and density \(\rho\) per unit length when unstretched lies in the form of a semicircle of radius \(a\) on the upper half of a smooth circular cylinder whose axis is horizontal. If \(T\) is the tension of the string at a point whose radius makes an angle \(\phi\) with the upward vertical, the angle to the same point of the string if lying unstretched symmetrically across the cylinder being \(\theta\), shew that \[ \frac{dT}{d\theta} + g\rho a \sin\phi = 0,\] where \[ T = \lambda \left(\frac{d\phi}{d\theta}-1\right). \] If \(\lambda = 2g\rho a\), deduce that \[ T = \lambda\left(\sqrt{2}\cos\frac{\phi}{2}-1\right), \] and that the unstretched length of the string is \(2a\sqrt{2}\log(\sqrt{2}+1)\).
Two particles, each of weight \(W\), are joined by a light elastic string of natural length \(l\) and modulus of elasticity \(W\). They are held on a rough plane inclined at \(60^\circ\) to the horizontal at a distance \(d(>l)\) apart, so that the string lies along a line of greatest slope. If the coefficient of friction between each particle and the plane is \(\sqrt{3}/2\), and the particles are released simultaneously, examine whether, and if so how, equilibrium is broken in the cases (i) \(d=\frac{3}{2}l\); (ii) \(d=\frac{5}{2}l\).
A particle of mass \(m\) is slightly disturbed from rest at the highest point of a smooth uniform hemisphere of radius \(a\) and mass \(M\) whose base is free to move on a smooth horizontal plane. Shew that, while the particle is in contact with the hemisphere, it describes an elliptic path in space, and find an equation in \(\cos\alpha\) to determine the angle \(\alpha\) which the radius to the particle makes with the vertical when the particle leaves the sphere. By sketching the graph \(y=x^3-3kx+2k\), where \(k>1\), or otherwise, shew that this equation in \(\cos\alpha\) has a unique root between \(0\) and \(1\), and shew that, whatever the ratio of \(M/m\), the particle leaves the hemisphere before the radius to it makes an angle \(\cos^{-1}(\frac{2}{3})\) with the vertical.
Find expressions for the tangential and normal components of the acceleration of a particle moving in a plane curve. A thin tube is in the form of the curve whose coordinates are given parametrically by the equations \(x=a(\theta+\sin\theta)\), \(y=a(1-\cos\theta)\), where the line \(y=0\) is horizontal and the line \(x=0\) is vertically upwards. A particle of mass \(m\) is joined by a light elastic string, of natural length \(a\sqrt{2}\) and modulus of elasticity \(mg/2\sqrt{2}\), lying inside the tube, to the point given by \(\theta=\frac{\pi}{2}\). It is held at rest at the point given by \(\theta=0\), and then released. Prove that the particle comes to rest again just as the string becomes slack. Prove also that the pressure on the particle due to the curve acts upwards throughout the motion, and find its value at the point whose parameter is \(\theta\).
A rod of mass \(M\) is free to rotate in a vertical plane about a fixed point \(O\). The moment of inertia of the rod about \(O\) is \(I\), and the distance of the centre of gravity from \(O\) is \(h\). When the rod makes an angle \(\alpha\) with the downward vertical, its angular velocity is \(\omega\). Determine the horizontal and vertical components of the reaction on the hinge when the rod makes an angle \(\theta\) with the downward vertical. Shew that these two components cannot vanish simultaneously in a position in which the rod is not vertical, unless \(I=Mh^2\) and the angular velocity \(\omega\) satisfies the inequalities \[ \frac{2g\cos\alpha}{h} < \omega^2 < \frac{g(2\cos\alpha+3)}{h}. \] If the two components vanish simultaneously when the rod is vertical, prove that \[ \omega^2 = \frac{g}{h}\left[\frac{4Mh^2\cos^2\frac{\alpha}{2}}{I} + 1\right]. \]
A square framework formed of four equal uniform rods each of weight \(W\) is hung up by one corner. The rods are freely jointed at each corner, and a weight \(W\) is suspended from each of the three lower corners. The shape of the square is preserved by means of a light rod along the horizontal diagonal. Prove that the thrust in this rod is \(4W\).
A rod is in equilibrium resting over the rim of a smooth hemispherical bowl fixed with its rim horizontal. One end of the rod rests on the curved surface. Show that the inclination, \(\theta\), of the rod to the horizontal is given by \(4r\cos 2\theta = l\cos\theta\), where \(l\) is the length of the rod and \(r\) the radius of the bowl. Show also that \(l\) must satisfy \[ \frac{2\sqrt{6}}{3}r < l < 4r. \]
A sash window of breadth \(a\), height \(b\), and weight \(W\) hangs in its frame with one of its cords broken. The remaining cord passes over its pulley and is attached to a counterpoise of weight \(W/2\). The window is slightly loose in its frame so that it makes contact only at two opposite corners. Show that the least value for the coefficient of friction between the window and the frame consistent with equilibrium is given by \(\mu=b/a\).