From a uniform circular disc or hole of diameter D, a circular disc or hole of diameter \(\frac{D}{4}\) from the center of disc is cut-off. Find: the centre of mass of the remaining portion.
Ans: \(\frac{D}{140}\) from the center of the disc and towards the left side of centre of the disc.
A rod PQ of mass M and length L is hinged at the end P. The rod is kept horizontal by a massless string tied to point Q to the celling. If the string is cut find the initial angular acceleration produced in the motion of the rod.
Ans: \(\alpha=\frac{3}{2}\frac{g}{L}\).
A disc of mass \(200kg\) and radius \(0.5m\) is rotating at a rate of 8 revolution per second. How much constant torque is required to stop the wheel in 11 rotations.
Ans: \(\tau=457.01N\).
A frame consists of a uniform circular ring of radius \(25cm\) and mass \(1.5kg\) and a uniform rod of length \(48cm\) and mass \(0.6kg\). The end A and B of the rod are attached to points on the circumference of the ring as shown below. Find the distance of the centre of mass of the frame from centre of the ring.
Ans: \(\left(0.02,0\right)\).
One end of a uniform rod of mass M and length L is supported by a frictionless hinge which can withstand a tension of \(1.75Mg\). The rod is free to rotate in vertical plane. To what maximum angle should the rod be rotated from vertical position so that when left, the hinge do not break?
Ans: \(\theta=60^{0}\).|
A tube of length L is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity \(\omega\). How much force is exerted by the liquid at the other end?
Ans: \(F=\frac{1}{2}M\omega^{2}L\) .
The moment of inertia of a body about a given axis is \(1.2kg-m^{2}\). Initially the body is at rest. In order to produce a rotational kinetic energy of \(1500J\), for how much durations, an acceleration of \(25\frac{rad}{sec}\) must be applied about the axis?
Ans: \(t=2s\).
A ring of mass \(0.3kg\) and radius \(0.1m\) & a solid cylinder of mass \(0.4kg\) and of same radius are given the same kinetic energy and released simultaneously on a flat horizontal surface such that they begins to roll as soon as released towards a wall which is at the same distance from the ring and cylinder. The rolling friction in both cases is negligible. The cylinder will reach the wall first. State weather the statement is true or false.
Ans: So the statement id true.
An isolated particle of mass m is moving in a horizontal plane \(\left(x-y\right)\) along the x-axis at a certain height above the ground. If suddenly explodes into two fragments of masses \(\frac{m}{4}\) and \(\frac{3m}{4}\). An instant later, the smaller fragment is at \(y=+15cm\). What is the position of larger fragment at this instant?
Ans: \(-5cm\).
A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligible small push is given to it. Calculate: the horizontal velocity of the cylindrical part of the carpet when its radius reduces to \(\frac{R}{2}\).
Ans: \(v=\sqrt{\frac{14}{3}gR}\).
A particle pf mass m is moving in a circular path of constant radius r such that its centrifugal acceleration \(a_{c}\) varies with time as; \(a_{c}=k^{2}rt^{2}\). Where; k is a constant. Calculate: the power delivered to the particle by the forces acting on it.
Ans: \(P=k^{2}.mr^{2}t\) .
A circular cylinder has an inextensible string wrapped around it as shown in below figure. What is the linear acceleration of the cylinder when released?
Ans: \(a=\frac{2}{3}g\).
Toque of equal magnitudes are applied on a hollow cylinder and sold solid sphere, both having same masses and same radius. The cylinder is free to rotate its standard axis of the symmetry and the spheres is free to rotate about an axis passing through its cemtre. Which of the two will acquire a greater angular speed after a given time?
Ans: \(a_{sphere}=a_{cylinder}\).
A thin wire of length L and uniform linear mass density \(\rho\) is bent into a circular loop with centre at O as shown in below figure. What is the moment of inertia of the loop about its axis XX’? Ans: \(I_{XX’}=\frac{3\rho L^{2}}{8\pi^{2}]\).
From a circular disc of radius R and mass 9M, a small disc of radius \(\frac{R}{3}\) is removed from the disc as shown in below figure. Find, the moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through the point O. Ans: \(I=4MR^{2}\).
A circular plate of uniform thickness has a diameter of \(56cm\). A circular portion of diameter of \(42cm\) is removed from one edge of the plate as shown in below figure. Find the position of the centre of mass of remaining portion. Ans: \(x=-9cm\).
Initial angular velocity of a circular disc of mass M is \(\omega_{1}\). Then two small spheres of mass m are attached gently to two diametrically opposite points on the edge of the disc. What will be the final angular velocity of the disc?
