Evaluate the integrals \[ \int \log x\,dx, \quad \int \frac{x^3\,dx}{\sqrt{x-1}}, \quad \int_0^{\frac{1}{2}\pi} \sin^2x\cos^2x\,dx. \]
Two equal parabolas of latus rectum \(4a\) have a common focus. Shew, by integration or otherwise, that if \(\alpha\) is the inclination of their axes, the area common to both is \(\dfrac{16}{3}a^2\text{cosec}^3\dfrac{\alpha}{2}\).
If four equal forces acting at a point are in equilibrium shew that they must consist of two pairs of opposite forces. \(A, B, C, D\) are four smooth holes in a horizontal table, \(ABCD\) being a convex quadrilateral, and four weights \(P, Q, P, Q\), are suspended below the table by strings passing up through \(A, B, C, D\) respectively, and all joined to a small smooth ring resting on the table. Shew that in equilibrium the ring must be at the intersection of \(AC\) and \(BD\). Examine whether this is true in the case of a crossed or re-entrant quadrilateral.
Three masses \(m_1, m_2, m_3\) are attached to three points \(A, B, C\) of a weightless string; \(m_1\) rests on a rough horizontal table, the part \(BC\) of the string hangs over a smooth peg \(P\) whose height above the table is greater than \(AB\), and \(m_2\) is lifted off the table. Shew that if \[ m_2 < m_3 < m_1 + m_2 \] the least angles \((\alpha, \beta)\) which the portions of string \(AB, BP\) can make with the horizontal are given by \[ \frac{\cos(\beta-\epsilon)}{\sin\epsilon} = \frac{m_2+m_1}{m_3} \quad \text{and} \quad \frac{\sin(\beta-\alpha)}{\cos\alpha} = \frac{m_2}{m_3}, \] where \(\epsilon\) is the angle of friction at the table.
\(AB\) represents the piston-rod of the fixed cylinder of a steam-engine, and \(CD\) is a crank turning an axle \(D\), \(BC\) being a connecting-rod. \(DE\) is drawn perpendicular to \(AB\) meeting \(BC\) in \(E\), and \(CF\) is the perpendicular from \(C\) on \(AB\); \(CG\) is perpendicular to \(BC\). [A diagram illustrates the described mechanism.] Shew that if the thrust in \(AB\) is given,
Shew that in a position of equilibrium the potential energy of a system has a maximum or minimum value. The lower ends of three identical vertical springs of length \(l\) and large modulus of elasticity \(\lambda\) are attached to three fixed points at equal distances \(a\) apart in a horizontal line. A bar of mass \(M\) is placed across their upper ends and attached to them in a position in which its centre of gravity is at a distance \(c\) from the middle spring. Find the potential energy when the middle spring is compressed a distance \(x\), and the rod makes a small angle \(\theta\) with the horizontal, and hence find the position of equilibrium and for what values of \(c/a\) one of the end springs becomes extended.
Shew how to find the resultant of any number of parallel forces acting at points of a plane, their lines of action being inclined to the plane, shewing that the point of the plane through which the resultant acts is independent of the direction of the forces. What is the bearing of this on what is known as the centre of gravity of a lamina?
An engine working at the steady rate of 600 horse power pulls a train of 250 tons up a hill with a slope of 1 in 140. If the resistances to motion, other than gravity, are equivalent to 17 lbs. weight per ton find (i) the acceleration of the train when its velocity is 15 miles per hour and (ii) the steady rate at which the engine can pull the train up this slope.
Prove that the free path of a particle moving under gravity is a parabola. A particle is projected from a point at a height \(h_0\) above the level of the ground and its height is \(h_1\) when it is at horizontal distances \(b\) and \(2b\) from its point of projection. Prove that the velocity with which it reaches the ground is \(v\), where \[ v^2 = g\left(\frac{4b^2}{4h_1-h_0} + 9h_1-h_0\right). \]
A particle describes simple harmonic motion with \(n\) complete vibrations per minute, being projected with velocity \(v\) feet per second from a point distant \(c\) feet from the central point of its motion. Find the amplitude of the motion. A body of mass 3 lbs. is attached to one end of a light elastic string whose other end is fixed and the string is such that a force of 1 lb. weight stretches it 1 inch. The body is projected downwards with a velocity of 8 feet per second from a position in which the string is vertical and at its full unstretched length. Shew that, taking \(g\) as 32 feet per sec. per sec., the body falls 12 inches and that the time of fall is \[ \frac{1}{8\sqrt{2}}\left(\frac{\pi}{2} + \sin^{-1}\frac{1}{3}\right) \text{ seconds}. \]