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
If the ice at the poles melts and flows towards the equator, how will it affect the duration of day-night?
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 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 ___________ .
In the previous question, the smallest kinetic energy at
the bottom of the incline will be achieved by ___________ .
A string of negligible thickness is wrapped several times around a cylinder kept on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder towards him (see the following figure). There is no slipping anywhere. The length of the string passed through the hand of the man while the cylinder reaches his hands is _________ .
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 joining two of the particles.
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.
Particles of masses 1 g, 2 g, 3 g, .........., 100 g are kept at the marks 1 cm, 2 cm, 3 cm, ..........., 100 cm respectively on a metre scale. Find the moment of inertia of the system of particles about a perpendicular bisector of the metre scale.
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.
A sphere of mass m rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed \[\nu.\]
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.
