Advertisements
Advertisements
प्रश्न
Two blocks of masses 400 g and 200 g are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia \[1 \cdot 6 \times {10}^{- 4} kg - m^2\] and a radius 2⋅0 cm, Find (a) the kinetic energy of the system as the 400 g block falls through 50 cm, (b) the speed of the blocks at this instant.
Advertisements
उत्तर
From the free body diagram, we have
\[0 . 4g - T_1 = 0 . 4a ..........(1)\]
\[ T_2 - 0 . 2g = 0 . 2a ..........(2)\]
\[\left( T_1 - T_2 \right) r = \frac{la}{r} ...........(3)\]
On solving the above equations, we get
\[a = \frac{\left( 0 . 4 - 0 . 2 \right) g}{\left( 0 . 4 + 0 . 2 + \frac{1 . 6}{0 . 4} \right)} = \frac{g}{5}\]
On solving the (b) part of the question first, we have
Speed of the blocks = \[v = \sqrt{\left( 2ah \right)} = \sqrt{2 \times \frac{g}{5} \times 0 . 5} = \sqrt{\left( \frac{9 . 8}{5} \right)} = 1 . 4 m/s\]
(a) Total kinetic energy of the system
\[= \frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2 + \frac{1}{2}I \omega^2 \]
\[ = \frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2 + \frac{1}{2}I \left( \frac{v}{r} \right)^2 \]
\[ = \left( \frac{1}{2} \times 0 . 4 \times \left( 1 . 4 \right)^2 \right) + \left( \frac{1}{2} \times 0 . 2 \times \left( 1 . 4 \right)^2 \right) + \left( \frac{1}{2} \times 1 . 6 \times {10}^{- 4} \times \left( \frac{1 . 4}{2 \times {10}^{- 2}} \right)^2 \right)\]
\[ = \left( 0 . 2 + 0 . 1 + 0 . 2 \right) \left( 1 . 4 \right)^2 \]
\[ = 0 . 5 \times 1 . 96 = 0 . 98 \text{ joule}\]
APPEARS IN
संबंधित प्रश्न
Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be MR2/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge
Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?
A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?
Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω1 and ω2 are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ω1 ≠ ω2.
The moment of inertia of a uniform semicircular wire of mass M and radius r about a line perpendicular to the plane of the wire through the centre is ___________ .
Let I1 an I2 be the moments of inertia of two bodies of identical geometrical shape, the first made of aluminium and the second of iron.
Let IA and IB be moments of inertia of a body about two axes A and B respectively. The axis A passes through the centre of mass of the body but B does not.
Suppose the smaller pulley of the previous problem has its radius 5⋅0 cm and moment of inertia 0⋅10 kg-m2. Find the tension in the part of the string joining the pulleys.
The descending pulley shown in the following figure has a radius 20 cm and moment of inertia 0⋅20 kg-m2. The fixed pulley is light and the horizontal plane frictionless. Find the acceleration of the block if its mass is 1⋅0 kg.
Solve the previous problem if the friction coefficient between the 2⋅0 kg block and the plane below it is 0⋅5 and the plane below the 4⋅0 kg block is frictionless.
A uniform metre stick of mass 200 g is suspended from the ceiling thorough two vertical strings of equal lengths fixed at the ends. A small object of mass 20 g is placed on the stick at a distance of 70 cm from the left end. Find the tensions in the two strings.
A diver having a moment of inertia of 6⋅0 kg-m2 about an axis thorough its centre of mass rotates at an angular speed of 2 rad/s about this axis. If he folds his hands and feet to decrease the moment of inertia to 5⋅0 kg-m2, what will be the new angular speed?
Four bodies of masses 2 kg, 3 kg, 4 kg and 5 kg are placed at points A, B, C, and D respectively of a square ABCD of side 1 metre. The radius of gyration of the system about an axis passing through A and perpendicular to plane is
A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind (Figure). The moment of inertia about the z-axis is then ______.
Moment of inertia (M.I.) of four bodies, having same mass and radius, are reported as :
I1 = M.I. of thin circular ring about its diameter,
I2 = M.I. of circular disc about an axis perpendicular to disc and going through the centre,
I3 = M.I. of solid cylinder about its axis and
I4 = M.I. of solid sphere about its diameter.
Then -
Four equal masses, m each are placed at the corners of a square of length (l) as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be ______.
A thin circular plate of mass M and radius R has its density varying as ρ(r) = ρ0r with ρ0 as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = a MR2. The value of the coefficient a is ______.
