10273 problems found
A cylinder \(A\) of radius \(a\) is eccentrically loaded so that its centre of gravity \(G\) is distant \(h\) from its axis. It rests in equilibrium on a fixed cylinder \(B\) of radius \(b\), with the highest generator of \(B\) along the lowest generator of \(A\) and \(G\) vertically below the axis of \(A\). Assuming the surfaces rough enough to prevent slipping, investigate the condition that the equilibrium may be stable.
A point moves with uniform acceleration on a straight line. Shew that the time-average of the velocity in any interval is equal (i) to the arithmetic mean of the velocities at the beginning and end of the interval, (ii) to the velocity at the middle of the interval. \par If the point travels 24 feet and 36 feet in two successive intervals of 2 seconds and 4 seconds respectively, determine how much farther it will travel, and how much longer it will take, before coming to rest.
A wedge of mass \(m\) and angle \(\alpha\) is at rest on a table. A mass \(2m\) is placed on the face of the wedge and slides downwards under gravity. Assuming the surfaces in contact to be smooth, find the acceleration of the wedge and the reaction of the table on the wedge.
Define work and power. \par The engine of a car of mass \(17\frac{1}{2}\) cwt. works at a constant rate of 10 horse-power, the resistance to motion being proportional to the speed. The maximum speed is 60 miles per hour. Prove that the car will increase its speed from 30 miles per hour to 45 miles per hour in about a quarter of a mile.
A particle is projected from a given point \(A\) so as to pass through a given point \(B\) where the distance \(AB\) is \(l\) and the line \(AB\) makes an angle \(\theta\) with the horizontal. Prove that the least possible velocity of projection is \[ \sqrt{\{gl(1+\sin\theta)\}}. \] Suppose now that the particle instead of moving freely in space is projected along the surface of a smooth plane, which contains \(A\) and \(B\) and is inclined at an angle \(\phi\) to the horizontal. Prove that the least velocity of projection for the particle to pass through \(B\) is \[ \sqrt{\{gl(\sin\phi + \sin\theta)\}}. \]
On a smooth plane inclined at an angle \(\alpha\) to the horizontal a particle is lying at rest attached to a fixed point above the plane by an inextensible string making an acute angle \(\beta\) with the plane. Prove that it is possible to project the particle so that it describes a complete circle on the plane if \(\cot\alpha \ge 6\tan\beta\).
Establish the equation of motion of a rigid body which is rotating about a fixed axis under the action of forces. \par A body consists of two uniform rods each of mass \(M\) and of length \(2a\) rigidly fixed at right angles at their middle points. It is suspended by an end of one of the rods and is free to move in its own plane. Find the period of small oscillations about the position of stable equilibrium.
Solve the simultaneous equations: \begin{align*} x(y+z-x) &= a^2, \\ y(z+x-y) &= b^2, \\ z(x+y-z) &= c^2. \end{align*}
If one root of the equation \(x^3+ax+b=0\) is twice the difference of the other two, prove that one root is \(\frac{13b}{3a}\).
If \(p > q-2\), find the number of ways in which \(p\) positive signs and \(q\) negative signs can be placed in a row so that no two negative signs shall be together.