A set of steps smoothly hinged at the top is placed with the side containing the steps making an angle of \(45^\circ\) with the horizontal and the back making an angle of \(60^\circ\) with the horizontal. The bottom of the back presses against a stop so that it cannot slip. A man climbs the steps, and when he is half-way up, the steps slip. Show that neglecting the mass of the steps, the coefficient of friction between the steps and the floor is \((2-\sqrt{3})\).
An anemometer consists of 4 brass cylindrical bars each of length 1 ft. and of radius \(\frac{1}{4}\) inch. These are fastened at right angles to a vertical shaft of negligible cross-section. The end of each bar is attached to the rim of a hemispherical brass cup bounded by two concentric hemispheres of \(3\frac{1}{2}\) and 4 inches diameter, the centre of the spheres lying on the axis of the corresponding bar. If the anemometer is making 40 revolutions a minute, calculate its kinetic energy. (Assume the density of the brass to be 8.4.)
A steamer moving with constant speed, \(v\), relative to the water passes round a lightship anchored in a tideway, keeping the lightship always dead abeam. Show that the path of the steamer is an ellipse whose minor axis is in the direction of the tidal current and whose eccentricity is \(u/v\). (\(u\) is the speed of the tide and we assume \(u < v\).)
A train slows down on entering a station and stops with a slight jerk. Discuss the motion of a sliding door in the side of the train, stating clearly what conditions you assume.
An engine driver of a train at rest observes a truck moving towards him down an incline of 1 in 60 at a distance of half a mile. He immediately starts his train away from the truck at a constant acceleration of 0.5 ft./sec.\(^2\). If the truck just catches the train find its velocity when first observed. Assume that friction opposing the truck's motion is 14 lbs. weight per ton.
A light elastic string of unstretched length \(l\) hangs vertically supporting a mass \(m\) and is extended by a length \(b\). A mass \(m'\) is taken from \(m\) and the remaining mass is set free to oscillate. Find the greatest value of \(m'\) such that the string always remains taut in the subsequent motion. As in the previous case, a mass \(m_1(
According to Hesiod the anvil of Vulcan would take 9 days and 9 nights to fall from the Earth to the realms of Hades. Placing Hades at the centre of the Earth and assuming that the acceleration downwards varies directly as the distance from the centre (and is 32 ft./sec.\(^2\) at the Earth's surface) show that Hesiod's figures would give a value of about \(15 \times 10^6\) miles for the Earth's radius.
\(D, E, F\) are the feet of the perpendiculars from the vertices on the opposite sides of the triangle \(ABC\). Prove that if \(DE\) meets \(AB\) in \(R\) and \(RQ\), drawn parallel to \(BC\), meets \(AC\) in \(Q\), then \(FQ\) will bisect \(BC\).
Prove that the inverse of a circle is a circle or a straight line and find in each case the position of the point into which the centre of the circle inverts. Two circles \(A\) and \(B\) are given. A variable circle \(C\) passes through the centre of \(A\) and touches \(B\). Prove that the common chord of \(A\) and \(C\) touches a fixed circle coaxal with \(A\) and \(B\).
Prove that the points of intersection of pairs of opposite sides of a hexagon inscribed in a conic are collinear. Prove that if a variable conic through four fixed points \(A, B, C, D\) meets fixed lines through \(A\) and \(B\) in \(X\) and \(Y\), then \(XY\) passes through a fixed point on \(CD\).