10273 problems found
Two identical rectangular blocks each of mass \(m\) rest on a horizontal table with two faces in contact. A wedge consisting of an isosceles triangular prism of mass \(M\) and angle \(2\alpha\) has its vertex inserted between them. If the table and the faces of the wedge are smooth, shew that the wedge will descend under its own weight with acceleration \[ \frac{Mg}{M+2m\tan^2\alpha}. \] If the table and the faces of the wedge are rough, but not sufficiently so to prevent sliding, shew that the acceleration is \[ g \frac{M\cot(\alpha+\epsilon) - \mu(M+2m)}{M\cot(\alpha+\epsilon) - \mu M + 2m \tan\alpha}, \] where \(\epsilon\) is the angle of friction at the face of the wedge and \(\mu\) is the coefficient of friction between the blocks and the table. If the height of each block is \(h\) and the length of each in the direction of motion is \(2a\), find the greatest allowable value of \(h/a\) in order that the motion may take place without the blocks tipping.
A uniform rod of mass \(M\) and length \(2l\) is freely pivoted about its centre so that it can rotate in a vertical plane under gravity. A particle of mass \(m\) is attached by a string of length \(a\) to one end of the rod. It is found that it is possible for the system to oscillate about its equilibrium position so that the rod and string remain in the same plane and make equal and opposite angles with the vertical. Shew that \(Ml = 6(a-l)m\), and find the period of the oscillation.
A sphere collides simultaneously with two other spheres which are at rest and in contact; all three spheres are smooth and equal to one another. Assuming that for each point of impact the relative velocity along the normal after the impulse is \(-e\) times the relative velocity along the normal before the impulse, shew that \[ \tan\theta' = \frac{2-e}{2-3e}\tan\theta, \] where \(\theta, \theta'\) are the angles that the velocity of the first sphere makes with the perpendicular to the line of centres of the other two spheres before and after contact.
State the conditions under which a body will remain in equilibrium when acted on by three non-parallel forces. A uniform pole \(AB\), 13 ft. long, is suspended by ropes \(CA\) and \(DB\) from two points \(C\) and \(D\) which are at the same level and less than 12 ft. apart, and hangs so that \(B\) is 4 ft. and \(A\) 9 ft. below the line \(CD\). If the rope \(DB\) is 5 ft. long, find the length of the rope \(CA\), and the ratio of the tensions in the two ropes.
A particle of mass \(m\) rests on a plane inclined at an angle \(\alpha\) to the horizontal, and the angle of friction between the particle and the plane is \(\phi\). A horizontal force \(P\), making an angle \(\beta\) with vertical planes containing lines of greatest slope, is just sufficient to move the particle horizontally. Prove that \[ \tan\beta = \frac{\tan\phi}{\sin\alpha} \quad \text{and} \quad P = \frac{mg\sqrt{\sin^2\alpha+\tan^2\phi}}{\cos\alpha}. \]
The cantilever frame shown in Fig. 1 is built up of light rods and freely hinged throughout. Find the thrusts or tensions in the rods due to a load of 5 tons at \(A\).
A light flexible belt passes over a fixed pulley and is in contact with it for an angle \(\theta\) of the pulley circumference. The tensions on the free ends of the belt are \(T_1\) and \(T_2\) and the coefficient of friction between belt and pulley is \(\mu\). By considering the equilibrium of an element of belt embracing an angle \(\delta\theta\) of the pulley circumference, prove that the belt will be on the point of slipping if \(T_1/T_2 = e^{\mu\theta}\).
The surface bounded by the parabola \(x^2=4ay\), the axis of \(y\) and the line joining the points \((0,h)\) and \((2\sqrt{ah}, h)\) is revolved about the axis of \(y\) to form a solid of revolution. Prove that a body of this shape, made of uniform material, will stand in stable equilibrium on a horizontal surface with its axis vertical if \(h \le 2a(1+\sqrt{2})\). Find the angle to which the axis will be tilted if \(h=5a\) and the body rests in equilibrium with the parabolic surface in contact with the horizontal plane.
A flywheel of mass 80 lb. is suspended with its axis vertical by three vertical cords placed equidistant from the axis and at the corners of an equilateral triangle of side 1 ft., and each rope is 5 ft. long.
A car has mass \(M\) and is subjected to a constant net propulsive force \(P\), while wind effects produce a force \(aV^2\), \(V\) being the velocity of the car relative to the air, and \(a\) being a constant. The car starts from rest with a following wind of velocity \(U\). Find the time taken in accelerating to a speed equal to the maximum obtainable in still air.