Calculating Moment of Inertia for Pair of Solid Spheres
The problem asks for the moment of inertia of a pair of solid spheres about a tangent passing through their point of contact. We are given the properties of two solid spheres:
- Sphere 1: mass $m_1 = 5$ kg, radius $r_1 = 10$ cm $= 0.1$ m.
- Sphere 2: mass $m_2 = 10$ kg, radius $r_2 = 20$ cm $= 0.2$ m.
The moment of inertia of a solid sphere about an axis passing through its center and perpendicular to a diameter (i.e., about a diameter) is given by the formula: $I_{cm} = \frac{2}{5}MR^2$
To find the total moment of inertia about the specified tangent axis, we calculate the moment of inertia for each sphere using this formula and sum them up.
Moment of Inertia Calculation: Sphere 1
Using the formula $I = \frac{2}{5}MR^2$ for Sphere 1:
$ I_1 = \frac{2}{5} m_1 r_1^2 $ Substitute the values: $ I_1 = \frac{2}{5} \times (5 \text{ kg}) \times (0.1 \text{ m})^2 $ $ I_1 = \frac{2}{5} \times 5 \times 0.01 $ $ I_1 = 2 \times 0.01 $ $ I_1 = 0.02 \text{ kg.m}^2 $Moment of Inertia Calculation: Sphere 2
Using the formula $I = \frac{2}{5}MR^2$ for Sphere 2:
$ I_2 = \frac{2}{5} m_2 r_2^2 $ Substitute the values: $ I_2 = \frac{2}{5} \times (10 \text{ kg}) \times (0.2 \text{ m})^2 $ $ I_2 = \frac{2}{5} \times 10 \times 0.04 $ $ I_2 = 4 \times 0.04 $ $ I_2 = 0.16 \text{ kg.m}^2 $Total Moment of Inertia
The total moment of inertia of the pair of spheres about the tangent passing through the point of contact is the sum of the individual moments of inertia:
$ I_{total} = I_1 + I_2 $ $ I_{total} = 0.02 \text{ kg.m}^2 + 0.16 \text{ kg.m}^2 $ $ I_{total} = 0.18 \text{ kg.m}^2 $Similar Questions
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)
- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- Given below are two statements :
Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II : The time period of revolution of the satellite is $T = 2\pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_e$ radius of earth and g acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below : - When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
- Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is ______ $\text{kg.m}^2$. ($g = 9.8 \text{ m/s}^2$)
Important Questions from Mechanics
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- Given below are two statements :
Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II : The time period of revolution of the satellite is $T = 2\pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_e$ radius of earth and g acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below : - When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
