A stream of particles moving at speed \(v\) falls upon a perfectly elastic plane reflecting surface at an angle of incidence \(\alpha\). If there are \(n\) particles, each of mass \(m\), per unit volume in the incident stream, calculate (i) the number of particles falling on unit area of the surface per unit time, (ii) the force per unit area needed to hold the surface in position. A gas is enclosed in a rectangular box. If there are \(N\) molecules per unit volume, each of mass \(m\), all moving with the same speed \(v\) in directions distributed uniformly in space, show that there are \(\frac{1}{2} N \sin \alpha d\alpha\) molecules per unit volume with velocities making angles lying between \(\alpha\) and \(\alpha + d\alpha\) with any given direction. Calculate the pressure exerted on the sides of the box, assuming them to be perfectly elastic.
\(AB, BC\) are two uniform rods of weights \(W, W'\), freely hinged to each other at \(B\) and freely hinged to points \(A\) and \(C\) in a horizontal plane. The rods stand in a vertical plane. The length of \(AB\) is 20 ft., of \(BC\) 15 ft. and of \(AC\) 10 ft. Find the magnitudes of the horizontal and vertical components of the reaction at \(B\), and the distance from \(C\) of the point at which the line of action of the resultant reaction meets \(AC\).
\(AB\) is a diameter of a solid uniform sphere of radius \(a\) and \(O\) is the centre. Find the distances from the centre of the sphere of the centres of gravity of the two parts into which the sphere is divided by a plane bisecting \(BO\) perpendicularly. Find also the position of the centre of gravity of the part of the sphere contained within the cone obtained by joining \(A\) to the circle in which the plane cuts the sphere.
Two identical uniform rectangular blocks of weight \(w\), height \(2h\), breadth \(2a\) and length \(l\), lie on a rough horizontal plane (coefficient of friction \(\mu\)) with their ends in the same planes. A wedge of the same length \(l\) whose section is an isosceles triangle of vertex angle \(2\alpha\) lies with its vertex downwards resting symmetrically on the blocks. The angle of friction between an edge of each block and a face of the wedge is \(\gamma\). A gradually increasing weight is placed symmetrically on the wedge. Shew that the blocks will slip or tip first according as \(\mu/\{\cot(\alpha+\gamma)-\mu\}\) is less or greater than \(a/2\{h\cot(\alpha+\gamma)-a\}\), provided both of these quantities are positive. What happens if either or both are negative?
The figure represents a roof-truss supported at \(A\) and \(F\). \(AF\) is horizontal, \(CD\) is vertical. Each of the members \(AB, BC, BD, CD, CE, DE, EF\) has the same length, 10 ft. Find graphically the stresses produced in the various members by a wind pressure on the roof represented by forces applied at \(A, B\) and \(C\) at right angles to \(AC\) of magnitudes 100 lb., 200 lb. and 100 lb. respectively. It may be assumed that the horizontal component of the total pressure is borne entirely by the bearing \(A\), so that the reaction at \(F\) is vertical. % Figure is a symmetric roof truss. A and F are supports. C is the peak. % Members are AC, CE, EF. From C, CD is a vertical strut down to D on AF. % From B on AC and E on CF, struts BD and ED meet at D.
A table stands on four identical vertical legs on a horizontal plane, the feet of the legs forming a square \(ABCD\) of side \(a\). The vertical line drawn through the centre of gravity of the table and the load upon it meets the plane \(ABCD\) in a point \(O\) distant \(p\) from \(AB\) and \(q\) from \(DA\). The legs are slightly compressible so that the thrust in each leg is \(k\) times its compression. Shew that the pressures on the ground at \(A, B, C, D\) are in the ratios \[ 3a - 2(p+q) : a+2(q-p) : 2(p+q)-a : a-2(q-p), \] provided that these are all positive.
A ball is thrown from a point on the ground with velocity \(V\). Shew that, if it passes over the top of a wall of height \(h\) at horizontal distance \(a\), \(V^2\) must be greater than \(g\{h + \sqrt{(a^2+h^2)}\}\). If this condition is satisfied, find between what limits the direction of projection must lie.
A train moves from rest under a force \(P-kv^2\), \(k\) being a constant and \(v\) the velocity. Shew that the time taken to reach two-thirds of the greatest possible velocity is \(t_0 \log_e 5\), where \(t_0\) is the time that would be taken to reach the same velocity, if the force remained constant at the value \(P\), and that the distance moved in acquiring that velocity is \(s_0 \log_e \frac{9}{5}\), where \(s_0\) is the corresponding distance if the force remained constant.
A particle of mass \(m\) is attached by two elastic strings of different moduli of elasticity to two points \(A, B\) of a horizontal table. The unstretched lengths of the strings are \(a,b\) and the stretched lengths in the equilibrium position \(a', b'\). If the periods of small oscillations in the directions along \(AB\) and perpendicular to \(AB\) are \(2\pi/n_1\) and \(2\pi/n_2\) respectively, shew that \[ n_1^2 = \frac{1}{m}(\frac{\lambda_1}{a}+\frac{\lambda_2}{b}) \quad \text{This seems to be missing from the OCR, I will transcribe what is there.} \] shew that \[ n_2^2 = \frac{1}{m}(\frac{1}{a'} + \frac{1}{b'}) \quad \text{This seems to be missing from the OCR, I will transcribe what is there.} \] \[ n_1^2 = \dots \quad \frac{1}{a'} + \frac{1}{b'} \] \[ n_2^2 = \dots \quad \frac{1}{a'-a} + \frac{1}{b'-b} \] % The OCR is extremely garbled for this question. I'm transcribing the visible fragments. The formulas seem to be definitions for n1^2 and n2^2. % A reasonable guess would be: % \frac{n_1^2}{m} = \frac{\lambda_1}{a'-a} + \frac{\lambda_2}{b'-b} % \frac{n_2^2}{m} = \frac{T_1}{a'} + \frac{T_2}{b'} = \frac{\lambda_1(a'-a)}{a a'} + \frac{\lambda_2(b'-b)}{b b'} % But I will transcribe what is there: \[ n_2^{-2} = \frac{1}{a'} + \frac{1}{b'} \] \[ n_1^2 = \frac{1}{a'-a} + \frac{1}{b'-b} \] % This is almost certainly wrong, but it's what's printed.
A pile of mass \(M\) is driven into the ground by the impact of a mass \(m\) falling vertically on it. The resistance of the ground to the motion of the pile is a force of magnitude \(R\). If the mass \(m\) strikes with a velocity \(V\) and there is a coefficient of elasticity \(e\), shew that there is a second impact after a time \(2eMV/R\), provided that the pile does not come to rest before that time; shew also that the condition for this to happen is \[ R-Mg < \frac{(1+e)mMg}{2eM - (1-e)m}. \] Shew that if this is so the mass comes to rest on the pile in a time \(2eMV/R(1-e)\).