In a molecule of carbon monoxide the distance between the carbon atom and the oxygen atom is \(\left(1.2\times10^{-10}\right)m\). What will be the distance of carbon atom from the centre of mass of the carbon monoxide molecule? The mass of oxygen atom is \(1.3\times\) the mass of carbon atom.
If, three point masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) are situated at the vertices of an equilateral triangle having side \(a\). The what will be the co-ordinates of C.M of the mass system?
Ans: \(X_{c.m}=\frac{m_{2}a+m_{3}\frac{a}{2}}{m_{1}+m_{2}+m_{3}}\). and \(Y_{c.m}=\frac{m_{3}\sqrt{3}a}{2\left(m_{1}+m_{2}+m_{3}\right)}\).
A T-shaped object with dimensions as shown in below figure is lying on a smooth floor. A force of magnitude \(F\) is applied on the point \(P\) parallel to AB such that the object has only translational motion without rotation. Find: the distance of location of point \(P\) from point C.
Ans: \(y=\frac{4}{3}l\).
Two identical uniform rods of length \(l\) are joined to form L shaped frame as shown in below figure. Locate the position of the centre of mass of the frame?
A circular disc of radius R has a uniform thickness. a circular hole of diameter equal to the radius of the disc has been cut out as shown in below figure. Find: the centre of mass of remaining disc.
Ans: \(X_{c.m}=-\frac{R}{6}\).
Two blocks 1 and 2 of masses \(m_{1}\) and \(m_{2}\) connected by a weightless spring of stiffness K rest on a smooth horizontal plane. Block 2 is shifted through a small distance x to the left and released. Find: the velocity of the centre of mass of the system after block 1 breaks of the wall.
A circular plate of uniform thickness has a diameter of \(56cm\). A circular portion of diameter \(42cm\) is removed from one edge of the plate. Find: centre of mass of the remaining portion of circular plate(disc)?
Ans: \(9cm\).
Determine the co-ordinates of centre of mass of a right circular solid cone having radius R and height h.
Ans: \(\left(0,\frac{h}{4},0\right)\).
A thin rod of length L is lying along the x-axis with its ends at x=0 to x=L. It’s linear density (mass/length) varies with x as \(K\left[\frac{x}{L}\right]^{n}\), where: n can be zero or any positive number. Find: the position of center of mass of the rod and plot the graph showing the variation of the position of the center of mass with value of n.
Ans: \(X_{c.m}=\left(\frac{n+1}{n+2}\right)L\).
Two bodies of masses \(2kg\) and \(10kg\) have position vectors \(\left(3\hat{i}+2\hat{j}-\hat{k}\right)\) and \(\left(\hat{i}-\hat{j}+3\hat{k}\right)\) respectively. Find: the position vector and distance of C.M from the origin.
Ans:(i) \(\frac{1}{6}\left(8\hat{i}-3\hat{j}+14\hat{k}\right)\) and (ii) \(2.73Units\).
A \(2kg\) particle has a velocity of \(\left(2\hat{i}-\hat{j}\right)\frac{m}{s}\) and a \(3kg\) particle has a velocity of \(\left(\hat{i}+6\hat{j}\right)\frac{m}{s}\). Find: (i) Velocity of the centre of mass. (ii) Total momentum of the system of particle.
Ans:(i) \(\left(1.4\hat{i}+3.2\hat{j}\right)\frac{m}{s}\) and (ii) \(\left(7\hat{i}+16\hat{j}\right)kg.\frac{m}{s}\).
Two particles of masses \(100g\) and \(300g\) at a given time have positions \(\left(2\hat{i}+5\hat{j}+13\hat{k}\right)m\) and \(\left(-6\hat{i}+4\hat{j}-2\hat{k}\right)m\) respectively and velocities \(\left(10\hat{i}-7\hat{j}-3\hat{k}\right)\frac{m}{s}\) and \(\left(7\hat{i}-9\hat{j}+6\hat{k}\right)\frac{m}{s}\) respectively. Determine the instantaneous position and velocity of C.M.
Ans:(i) \(\frac{\left(-16\hat{i}+17\hat{j}+7\hat{k}\right)}{4}m\) and (ii) \(\frac{\left(31\hat{i}-34\hat{j}+15\hat{k}\right)}{4}\frac{m}{s}\).
Three pieces of iron of uniform thickness and mass \(m\), \(m\) and \(2m\) respectively are placed at three corners of a triangle having co-ordinates \(\left(2.5,1.5\right)\), \(\left(3.5,1.5\right)\) and \(\left(3,3\right)\) respectively. Find: The center of mass of the system?
Ans: \(\left(3,2.25\right)\).
Three particles each of mass \(2.5kg\) are located at the corners of a right angle triangle whose sides are \(2m\) and \(1.5m\) long as shown in figure. Locate the center of mass.
Ans: \(\left(1.33m,0.5m\right)\).
A square of side \(4cm\) and uniform thickness is divided into four equal squares as shown in figure. If, one of the equal square is cut off from the square then find the position of center of mass of the remaining portion from center O.
Ans: \(\frac{\sqrt{2}}{3}\).
Four particles of masses \(m\), \(2m\), \(3m\) and \(4m\) respectively are placed at the corners of the sphere of side a as shown in below figure. Find the position of center of mass?
