Advertisements
Advertisements
प्रश्न
Suppose the rod in the previous problem has a mass of 1 kg distributed uniformly over its length.
(a) Find the initial angular acceleration of the rod.
(b) Find the tension in the supports to the blocks of mass 2 kg and 5 kg.
Advertisements
उत्तर
Total moment of inertia of the system about the axis of rotation,
\[I_{net} = \left( m_1 r_1^2 + m_2 r_2^2 + \frac{m l^2}{12} \right)\]
m and l are the mass and length of the rod, respectively.
\[ \tau_{net} = F_1 r_1 - F_2 r_2 \]
\[\text{Also, }\tau_{net} = I_{net} \times \alpha\]
On equating the value of \[\tau_{net}\] and putting the value of `l_("net"),` we get
\[F_1 r_1 - F_2 r_2 = \left( m_1 r_1^2 + m_2 r_2^2 + \frac{m l^2}{12} \right) \times \alpha\]
\[\left( - 2 \times 10 \times 0 . 5 \right) + \left( 5 \times 10 \times 0 . 5 \right) = \left[ 5 \left( \frac{1}{2} \right)^2 + 2 \left( \frac{1}{2} \right)^2 + \frac{\left( 1 \right)^2}{12} \right] \alpha\]
\[\Rightarrow 15 = \left( 1 . 75 + 0 . 084 \right) \alpha\]
\[ \Rightarrow \alpha = \frac{1500}{\left( 175 + 8 . 4 \right)} = \frac{1500}{183 . 4}\]
\[= 8 . 1\text{ rad/s}^2 ........\left( g = 10 \right)\]
\[= 8 . 01\text{ rad/s}^2 ......\left(\text{if }g = 9 . 8 \right)\]
(b) From the free body diagram of the block of mass 2 kg,
\[T_1 - m_1 g = m_1 a\]
\[ \Rightarrow T_1 = 2 \left( a + g \right)\]
\[ = 2\left( \alpha r + g \right)........ \left(\text{using, }a = \alpha r \right)\]
\[ = 2\left( 8 \times 0 . 5 + 9 . 8 \right)\]
\[ \Rightarrow T_1 = 27 . 6 N\]
From the free body diagram of the block of mass 5 kg,
\[m_2 g - T_2 = m_2 a\]
\[ \Rightarrow T_2 = m_2 \left( g - a \right)\]
\[ = 5 \left( g - a \right) = 5 \left( 9 . 8 - 8 \times 0 . 5 \right)............\left( a = \alpha r \right)\]
\[= 5 \times 5 . 8 = 29 N\]
APPEARS IN
संबंधित प्रश्न
A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r is made from an iron plate of thickness t/4. The relation between the moments of inertia IA and IB is __________ .
A closed cylindrical tube containing some water (not filling the entire tube) lies in a horizontal plane. If the tube is rotated about a perpendicular bisector, the moment of inertia of water about the axis __________ .
A cubical block of mass M and edge a slides down a rough inclined plane of inclination θ with a uniform velocity. The torque of the normal force on the block about its centre has a magnitude
A thin circular ring of mass M and radius r is rotating about its axis with an angular speed ω. Two particles having mass m each are now attached at diametrically opposite points. The angular speed of the ring will become
The centre of a wheel rolling on a plane surface moves with a speed \[\nu_0\] A particle on the rim of the wheel at the same level as the centre will be moving at speed ___________ .
A wheel of radius 20 cm is pushed to move it on a rough horizontal surface. If is found to move through a distance of 60 cm on the road during the time it completes one revolution about the centre. Assume that the linear and the angular accelerations are uniform. The frictional force acting on the wheel by the surface is ______________________ .
A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of a smooth incline and released. Least time will be taken in reaching the bottom by _________ .
A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top on an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by ___________ .
Three particles, each of mass 200 g, are kept at the corners of an equilateral triangle of side 10 cm. Find the moment of inertial of the system about an axis passing through one of the particles and perpendicular to the plane of the particles.
Find the moment of inertia of a pair of spheres, each having a mass mass m and radius r, kept in contact about the tangent passing through the point of contact.
The radius of gyration of a uniform disc about a line perpendicular to the disc equals its radius. Find the distance of the line from the centre.
Find the moment of inertia of a uniform square plate of mass m and edge a about one of its diagonals.
The surface density (mass/area) of a circular disc of radius a depends on the distance from the centre as [rholeft( r right) = A + Br.] Find its moment of inertia about the line perpendicular to the plane of the disc thorough its centre.
Because of the friction between the water in oceans with the earth's surface the rotational kinetic energy of the earth is continuously decreasing. If the earth's angular speed decreases by 0⋅0016 rad/day in 100 years find the average torque of the friction on the earth. Radius of the earth is 6400 km and its mass is 6⋅0 × 1024 kg.
The following figure shows two blocks of mass m and M connected by a string passing over a pulley. The horizontal table over which the mass m slides is smooth. The pulley has a radius r and moment of inertia I about its axis and it can freely rotate about this axis. Find the acceleration of the mass M assuming that the string does not slip on the pulley.
A small spherical ball is released from a point at a height h on a rough track shown in the following figure. Assuming that it does not slip anywhere, find its linear speed when it rolls on the horizontal part of the track.
