Define the Centre of Mass of a system of particles and shew that the point is unique. A particle at \(O\) is subject to forces given in magnitude and direction by \[ k_1.\vec{OP_1}, k_2.\vec{OP_2}, \dots k_n.\vec{OP_n}, \] where \(P_1, P_2, \dots P_n\) are definite points. Shew how the resultant can be calculated by finding a centre of mass. A rigidly connected system of particles of masses \(m_1\) at \(O_1, m_2\) at \(O_2, \dots\) and \(m_l\) at \(O_l\) is subject to a system of forces such that the particle \(m_s\) at \(O_s\) is acted upon by a force \(m_s k_r . \vec{O_s P_r}\) towards \(P_r\), where \(r\) takes values 1 to \(n\), and \(s\) takes values 1 to \(l\). Shew that the resultant is given by \(MK . \vec{OP}\), where \(O\) is the centre of mass of \(m_1\) at \(O_1, \dots m_l\) at \(O_l\), \(P\) is the centre of mass of \(k_1\) at \(P_1, \dots k_n\) at \(P_n\), and where \(M=\sum m_s, K = \sum k_r\).
Two rough fixed parallel horizontal rails, with their common plane inclined \(\theta\) to the horizontal, support a hollow uniform semicircular cylinder resting with its axis parallel to the rails. The angle between the planes joining the rails and axis is \(2\alpha\), and \(\theta < \alpha < \frac{\pi}{2}\). [A diagram shows the cross-section of a semicircular cylinder. Its diameter is horizontal. Two points on the arc, symmetric about the vertical bisector, represent the points of contact with the rails. The angle subtended by these points at the center of the diameter is \(2\alpha\). The plane of the diameter is tilted at an angle \(\theta\) to the horizontal.] Shew that if the surfaces in contact are rough enough, the cylinder can rest in any position without turning about either rail if \[ \frac{\sin(\alpha-\theta)}{\cos(\alpha+\theta)} \ge \frac{2}{\pi+2}. \] Shew further, that if this condition is not satisfied, equilibrium will be broken by slipping if \(\lambda < \alpha - \theta\), \(\lambda\) being the angle of friction.
A flexible chain with line density \(w\) varying with distance by the relation \[ w = w_0 \sec^2 \frac{s}{a}, \] \(s\) being the distance measured from the mid-point, and \(w_0\) and \(a\) constants, hangs under gravity from two points at a variable distance apart on the same horizontal level. If the total length is \(\frac{\pi}{2}a\), shew that by suitable adjustment the chain can hang in an arc of a circle, and that in this case the least value of the tension in the chain is half its total weight.
A uniform heavy beam of length \(2a\) and weight \(2wa\) rests symmetrically on two supports on the same horizontal level. It is desired to arrange the supports symmetrically so that there is the least chance of the beam snapping. Find the distance between the supports for this arrangement, and shew that the beam must at least be capable of sustaining a bending moment of \(wa^2(3-2\sqrt{2})\).
A rigid smooth wire is held in a vertical plane in the form of a cycloid with vertex downwards. [The cycloid may be taken as \(x=a(\theta+\sin\theta), y=a(1-\cos\theta)\), with origin at vertex, and tangent there as \(x\) axis.] A bead of mass \(m\) rests at the vertex. Another bead of mass \(2m\) is released at a distance \(3a\) along the wire from the vertex and slides under gravity. Shew that immediately after impact (assumed perfectly elastic) the smaller bead will rise to the cusp, and the larger a distance \(a\) along the wire. Indicate the nature of the subsequent motion.
A smooth bead is released from a fixed point and allowed to slide under gravity on a smooth fixed wire in the form of part of a parabola with axis vertical and vertex downwards. The particle on leaving the lower end of the wire subsequently describes a parabola in space equal in size to the former. Find the ratio of the heights of the point of initial release and of leaving the wire above the directrix of the first parabola, and shew that immediately prior to leaving the wire the bead exerts a pressure on the wire twice that of the component normal to the wire of the weight of the bead.
A body of mass \(M\) is propelled on the horizontal by an engine of constant power \(R\). The motion is subject to a resistance \(Kv^3\), where \(v\) is the speed and \(K\) a constant. Shew that during motion starting with any initial speed, the speed will remain either greater than or less than a certain value \(C\). Shew that the time taken and the distance traversed in increasing the velocity from \(\frac{1}{2}C\) to \(\frac{3}{4}C\) are respectively: \[ \frac{M}{\sqrt[4]{RK^3}} \log\frac{1}{2}, \quad \frac{M}{\sqrt[4]{R^3K}} [\log\frac{5}{3} - 2\tan^{-1}\frac{1}{2}]. \] (Note: OCR is very poor for this question. The first term \(\log\frac{1}{2}\) is negative, which is impossible for time. This is likely a transcription of a more complex term. Similarly for the second term. Transcribing as seen from a clearer view: \[ \frac{M}{\sqrt{R}K} \log \frac{4}{3}, \quad \frac{M}{\sqrt[4]{R^3 K}}[\log \frac{5}{3} - \tan^{-1} \frac{1}{2}]. \] This still looks odd. Let's use the OCR from the prompt which seems to have been corrected: \[ \frac{M}{\sqrt[4]{R K^3}} \log \frac{4}{3}, \quad \frac{M}{R^{3/4} K^{1/4}} [\log\sqrt{2} - \tan^{-1}\frac{1}{2}]. \] Let's stick to a literal transcription of the image, despite potential errors in the question itself. \[ \frac{M}{\sqrt[4]{R K^3}} \log \frac{4}{3}, \quad \frac{M}{R^{3/4} K^{1/4}} [\log\frac{5}{3} - 2\tan^{-1}\frac{1}{2}]. \] The OCR for the prompt seems to be `1/4 VRK log 4/3` and `M / R^(3/4) K^(1/4) [log 5/3 - 2 tan^-1 1/2]`. The image seems to show `1/4 R^3 K` under the radical for the second term. It is very blurry. Let's go with the provided OCR.
A particle is projected vertically upwards with speed \(u\). The motion is subject to gravity and to a resistance per unit mass of \(Kg\) times the speed, \(K\) being a constant. Find the greatest height above the point of projection to which the particle will rise, and shew that the total distance travelled by the particle from projection until the speed is again \(u\) is given by \(\frac{1}{K^2g} \log\frac{1}{1-K^2u^2}\), where \(Ku<1\). What happens if \(Ku \ge 1\)?
A rough rigid wire rotates in a horizontal plane with constant angular velocity \(\omega\) about a vertical axis through a point \(O\) of itself. A bead, which can slide on the wire, is released from relative rest at a distance \(a\) from \(O\). Shew that at any time subsequently, the distance \(r\) of the bead from \(O\) satisfies the equation \[ \frac{d^2r}{dt^2} + 2\mu\omega \frac{dr}{dt} = \omega^2 r, \] \(\mu\) being the coefficient of friction. Prove that after a time \(t\), the velocity of the bead is \[ a\omega e^{-\mu\omega t} \left\{\cosh(n\omega t) + \frac{\mu}{n}\sinh(n\omega t)\right\}^{\frac{1}{2}}, \] (Note: The expression from the scan seems different from the OCR. Let's re-read the scan) It appears to be \(a\omega e^{-\mu\omega t} \{\cosh 2n\omega t + \frac{\mu}{n} \sinh 2n\omega t \}^{\frac{1}{2}}\). I will use this.
Two masses \(m_1\) and \(m_2\) are supported by a light inextensible string slung over a rough pulley of radius \(a\) which can turn freely about a fixed horizontal axis, its moment of inertia about the axis being \(I\). A mass \(m\) is rigidly attached to the pulley at a point distant \(h(