Explain and justify the use of the Polygon of Forces. Forces act in order along the sides of a regular \(n\)-sided polygon. Shew that
A right-angled girder consisting of two equal thin uniform heavy planks of width \(2l\) joined at one long edge of each, rests at an inclination \(\theta\) to the symmetrical position on a fixed rough circular cylinder of radius \(a\) with its edge parallel to the horizontal axis of the cylinder. The coefficient of friction between the planks and the cylinder is \(\mu = \tan\lambda\). Shew that equilibrium is possible if \[ \mu > \tan\theta \cdot \left(\frac{2a - l - l\mu^2}{2a}\right), \] provided \(4l > 2a > l\sec^2\lambda\), where \(\lambda\) is less than \(\frac{\pi}{4}\).
Two uniform smooth spheres of radii \(a, b\), weights \(w_1, w_2\), are joined by an inextensible light thread of length \(l\) attached at each end to the surface of a sphere. Shew that the system can rest in equilibrium with the spheres in contact each with a smooth plane (inclined \(\alpha, \beta\) to the downward vertical), the planes meeting in a horizontal ridge over which the thread passes. Shew further that the equilibrium is stable.
A heavy flexible chain, of length \(l\) and uniform weight \(w\) per unit length, hangs from one end under gravity in a uniform horizontal wind. Assuming that the drag on any portion of the chain due to the wind is in the direction of the wind and proportional to the projection of that portion on the vertical, shew that the chain assumes a plane curve, the inclination to the vertical of the tangent at the free end being \(\alpha\), where \(\sin\alpha\) is a root of \(x^2+\frac{x}{\kappa}-1=0\), where \(\kappa\) is a constant. Shew also that if the free end be at a depth \(h\) below the fixed end, the centroid of the chain is displaced a distance \(\frac{\kappa h^2}{2l}\) horizontally by the effect of the wind.
Define Shearing Force and Bending Moment in a beam subjected to stress. A horizontal straight light rigid beam \(AC\) as shown is supported in three sections \(AH, HJ, JC\), hinged freely at \(H, J\), by the equally spaced supports \(A, B, C\). Forces of 3 tons weight vertically downwards are applied at \(D\) and \(E\), the respective mid-points of \(AB, BC\). If \(HJ\) is bisected by \(B\) and is less than \(AB\), find the position of \(H\) and \(J\) to make the greatest bending moment in the beam take its least possible value, and shew that in that case the shearing force at either hinge is 2 tons weight.
An elastic particle is projected from a point on a rough fixed plane inclined at an angle \(\alpha\) to the horizontal. If the coefficients of restitution and impulsive friction be \(e\) and \(\mu\) respectively, and if the velocity of projection is \(u\) down the line of greatest slope and \(v\) perpendicular to the plane, shew that bouncing will cease after a time \(\frac{2v}{g\cos\alpha(1-e)}\), when the particle will be at a distance from the point of projection equal to \[ \frac{2v}{g\cos^2\alpha(1-e)^2}[u(1-e)\cos\alpha+v(\sin\alpha-\mu e\cos\alpha)]. \]
A smooth wire is bent in the form of a plane horizontal curve and constrained to rotate with constant angular velocity \(\omega\) about a vertical axis through any point \(O\) of its plane. Shew that the velocity relative to the wire of a smooth bead moving freely along it is given by \(v^2=\omega^2r^2+\text{constant}\), where \(r\) is the radius from \(O\) to the bead. Shew also that the normal reaction per unit mass on the bead is \[ \left(\frac{v^2}{\rho}+2v\omega+r\omega^2\sin\phi\right), \] where \(\rho\) is the radius of curvature at the point of the wire, and \(\phi\) the angle between the radius vector and the tangent to the wire.
A particle is projected vertically upwards in a resisting medium, the resistance per unit mass being \(\frac{g}{\kappa^2}v^2\), where \(v\) is the velocity. If the initial velocity be \(u\), shew that the greatest height above the point of projection reached is \(\frac{\kappa^2}{2g}\log_e\frac{u^2+\kappa^2}{\kappa^2}\). Shew also that the velocity downwards on regaining the point of projection is \[ \frac{\kappa u}{\sqrt{u^2+\kappa^2}}. \]
A particle of mass \(m\) is freely suspended by a light rigid wire of length \(l\) from a support of mass \(m\) which can move freely on a smooth horizontal rail. The system is started by a blow \(B\) parallel to the direction of the rail given to the particle. Shew that provided \(B<2m\sqrt{gl}\), the particle will not rise above a certain level below the rail. Shew also that when the inclination of the string to the vertical is \(\theta\), the velocity \(v\) of the support is given by \[ v = \frac{B}{2m} \pm \frac{1}{2}\cos\theta\sqrt{\frac{B^2}{m^2}-4gl(1-\cos\theta)}{2-\cos^2\theta}}, \] and explain the ambiguity of sign.
Two unequal masses \(M_1\) and \(M_2\) are joined by a light inextensible string slung over a heavy, rough and freely pivoted pulley whose moment of inertia is \(Mk^2\), and radius is \(a\). \(M_1>M_2\). Shew that if \(\mu\) is the coefficient of friction between the string and the pulley, slipping will occur when the system is released from rest unless \[ \mu > \frac{1}{\pi}\log_e\frac{M_1(2M_2+M\frac{k^2}{a^2})}{M_2(2M_1+M\frac{k^2}{a^2})}. \] If slipping does occur, shew that the acceleration of the masses is \[ \frac{g(M_1-M_2 e^{\mu\pi})}{M_1+M_2 e^{\mu\pi}}, \] and that the angular acceleration of the pulley is \[ \frac{M_1M_2(e^{\mu\pi}-1)}{2agMk^2(M_1+M_2e^{\mu\pi})}. \]