10273 problems found
A bullet of mass 1 oz. is fired into a block of wood of mass 20 lb. which is suspended by a long string. The bullet becomes embedded in the block and the centre of mass of block and bullet rises to a height of 12 in. above its original position. What is the velocity of the bullet? \par If the resistance to the penetration of the bullet is \(675 v x^{1/2}\) lb. wt., where \(v\) is the velocity and \(x\) the penetration, shew that the final value of \(x\) is about 1 ft. The movement of the block during penetration may be neglected.
If a particle is moving in a curve, \(v\) being its velocity and \(\psi\) the angle between the direction of motion and a fixed direction, shew that the components of acceleration along the tangent and normal are \(\dfrac{dv}{dt}\) and \(v\dfrac{d\psi}{dt}\). \par If a particle moves on a rough inclined plane, the coefficient of friction being \(\mu\) and the inclination of the plane to the horizontal \(\alpha\), shew that the motion satisfies the equations \begin{align*} \frac{dv}{dt} &= -\mu g \cos\alpha + g \sin\alpha \cos\psi, \\ v\frac{d\psi}{dt} &= -g\sin\alpha \sin\psi, \end{align*} \(\psi\) being measured from the direction down a line of greatest slope. Hence shew that \[ v \sin^\mu\psi = C \left(\tan\frac{\psi}{2}\right)^{\cot\alpha}, \] where C is a constant.
ABCD is a rhombus of freely hinged light rods each of length \(l\). It is pivoted at A at a fixed point and C is connected to a point E vertically below A at a distance \(b\) from A by a vertical spring of natural length \(a\) and modulus of elasticity \(\lambda\); the lengths \(a, b, l\) are such that \(b>a+2l\). Two particles of mass \(m\) are attached at B and D and the whole rotates about AE with angular velocity \(\omega\). \par Shew that, if \(\omega^2 > \dfrac{g}{l} + \dfrac{\lambda(b-a-2l)}{mal}\), there is a position of relative equilibrium in which the rods make with the vertical an angle whose cosine is \[ \frac{mga+\lambda(b-a)}{l(ma\omega^2+2\lambda)}. \]
Prove that, if a number of forces act on a rigid body, the sum of them is equal to the mass multiplied by the acceleration of the centre of mass. \par Prove also that for a lamina rotating about a fixed axis perpendicular to it the sum of the moments of the forces about the axis is equal to the moment of inertia about the axis multiplied by the angular acceleration. \par A uniform circular disc of mass \(m\) and radius \(r\) is suspended with its plane vertical by pegs passing through two small holes drilled perpendicular to the plane of the disc, each distant \(a\) from the centre of the disc and distant \(d\) from each other at the same level above the centre of the disc. If one peg suddenly collapses, shew that the horizontal reaction at the other peg becomes immediately \[ \frac{mgd\sqrt{4a^2-d^2}}{4a^2+2r^2}. \]
Prove that, if \(x > 0, 0 < p < 1\), then \[ (1+x)^p < 1+px. \] Hence show that, if \(a>0, b>0\) and if \(n\) is a positive integer, then \[ (a^n+b)^{1/n} < a + \frac{b}{na^{n-1}}. \] Show further by using the identity \[ (a^n+b) - a^n = b \] \[ (a^n+b)^{\frac{n-1}{n}} + a(a^n+b)^{\frac{n-2}{n}} + \dots + a^{n-1} \] or otherwise, that \[ a + \frac{b}{na^{n-1}+\frac{1}{2}(n-1)\frac{b}{a}} < (a^n+b)^{1/n}. \]
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Prove that if \(\alpha, \beta, \gamma\) are the roots of the equation \[ x^3 - 3px^2 - 3(1-p)x + 1 = 0, \] then \[ \beta(1-\gamma) = \gamma(1-\alpha) = \alpha(1-\beta) = 1 \] or \[ \beta(1-\alpha) = \gamma(1-\beta) = \alpha(1-\gamma) = 1. \] Show that \(\alpha, \beta, \gamma\) are real if \(p\) is real.
Prove that the radius of the nine-points circle of a triangle is half the radius of the circumcircle. \par The distances between the centres of the escribed circles of a triangle being \(\alpha, \beta, \gamma\), prove that \[ 4R = \frac{\alpha^2}{r_2+r_3} = \frac{\beta^2}{r_3+r_1} = \frac{\gamma^2}{r_1+r_2} = \frac{(r_2+r_3)(r_3+r_1)(r_1+r_2)}{r_2r_3+r_3r_1+r_1r_2} = \frac{\alpha\beta\gamma}{2\sqrt{\{\sigma(\sigma-\alpha)(\sigma-\beta)(\sigma-\gamma)\}}}, \] where \(2\sigma = \alpha+\beta+\gamma\).
\(y\) is the implicit function of two variables \(x, \alpha\) defined by the equation \[ y = x + x\phi(y). \] If \(u\) is a function of \(y\) and \(F(u)\) a function of \(u\), prove that \[ \frac{\partial}{\partial x}\left\{F'(u)\frac{\partial u}{\partial \alpha}\right\} = \frac{\partial}{\partial \alpha}\left\{F'(u)\frac{\partial u}{\partial x}\right\}, \] and that \[ \frac{\partial u}{\partial x} - \phi(y)\frac{\partial u}{\partial \alpha} = \frac{\partial}{\partial x}\left[\{\phi(y)\}^2 \frac{\partial u}{\partial \alpha}\right], \] and, generally, \[ \frac{\partial^n u}{\partial x^n} = \frac{\partial^{n-1}}{\partial x^{n-1}}\left[\{\phi(y)\}^n \frac{\partial u}{\partial \alpha}\right]. \]
If \(p(x)\) is a polynomial of the \(k\)th degree and if \[ H_n(x) = e^{p(x)}\frac{d^n e^{-p(x)}}{dx^n}, \] prove that \[ H_n(x) = q_{k-1}(x)H_{n-1}(x) + q_{k-2}(x)H_{n-2}(x) + \dots + q_0 H_{n-k}(x), \] where \(q_i(x)\) is a polynomial of the \(i\)th degree in \(x\). \par Find the actual equation for the case in which \(p(x)=x^2\).