Calculate: the moment of inertia and radius of gyration of a thin uniform rod about an axis passing through its centre and perpendicular to its length.
Ans: \(I=\frac{1}{12}ML^{2}\) and \(k=\frac{L}{2\sqrt{3}}\).
Estimate: the moment of inertia and radius of gyration of a thin uniform rod about an axis passing through its one end and perpendicular to its length.
Ans: \(I=\frac{1}{3}ML^{2}\) and \(k=\frac{L}{\sqrt{3}}\).
The masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) are located at the vertices of an equilateral triangle of length \(d\). What will be the moment of inertia of the system about an axis along the altitude of triangle passing through \(m_{1}\).
Four different masses of \(6kg\), \(4kg\), \(5kg\) and \(7kg\) are respectively located at four corners PQR and S of a square of side \(1m\) as shown in figure. Calculate: the moment of inertia of the system about, (i) An axis passing through the point of intersection of the diagonals and perpendicular to the plane of the square. (ii) The side PQ and (ii) The diagonal QS.
Ans:(i) \(11kg.m^{2}\) and (ii) \(12kg.m^{2}\) and \(5.5kg.m^{2}\).
Calculate: the ratio of radii of gyration of a circular ring and a disc of same radii and of same mass about the axis passing through their centres and perpendicular th their planes.
Ans: \(\frac{k_{1}}{k_{2}}=\frac{\sqrt{2}}{1}\).
A wheel is rotating at a rate of 1200rpm and its kinetic energy is \(10^{6}J\). Determine the M.I of the wheel about its axis of rotation.
Ans: \(I=126.64kg.m^{2}\).
To increase the speed of flywheel from \(0r.p.m\) to \(300r.p.m\) the energy of \(484J\) is spent. Find: the moment of inertia of wheel.
Ans: \(I=0.7kg.m^{2}\).
A solid sphere is rolling on a frictionless plane surface about its axis of symmetry. Find: the rotational energy and ratio of its rotational energy to its total energy.
Ans: \(K.E_{rotational}=\frac{1}{5}Mv^{2}\) and \(\frac{K.E_{rotational}}{T.E}=\frac{2}{7}\).
Four point masses \(m\), \(2m\), \(3m\) and \(4m\) lies at the four corners of a square of side \(4m\) as shown in figure. Find: the moment of inertia of the system of masses about (i) The diagonal BD. (ii) The diagonal AC. and (iii) An axis passing through A and parallel to diagonal BD.
Ans:(i) \(32m\), (ii) \(48m\) and (iii) \(144m\).
A metre scale AB is held vertically with its one end A on the floor and then allowed to fall. Find: the speed of the other end B when it strikes the floor, assume that the end on the floor does not slip.
Ans: \(5.4\frac{m}{s}\).
A circular disc X of radius R is made from an iron plate of thickness t and another disc Y of radius 4R is made from iron plate of thickness \(\frac{t}{4}\). Show that M.I of disc Y about an axis passing through its centre and perpendicular to its plane is 64 times the M.I of disc X about an axis passing through its centre and perpendicular to the planes of the discs.
What will be the rotational K.E of the earth about its own axis if the mass of earth is \(\left(6\times10^{24}\right)kg\) and radius \(\left(6.4\times10^{6}\right)m\).
A thin rod of length \(L\) and mass \(M\) is bent at its mid point into two halves so that the angle between them is \(90^{o}\). Calculate: the M.I of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by two halves or rod.
