Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
To solve for the moment of inertia of the remaining part of the disc after cutting out the shaded regions, we start by calculating the moment of inertia of the entire disc and then subtract the moments of inertia of the removed parts.
Step 1: Moment of Inertia of the Whole Disc
The moment of inertia of a uniform circular disc about its central axis is given by:
\( I_{\text{disc}} = \frac{1}{2} M r^2 \)
Step 2: Moment of Inertia of the Removed Sections
Each of the removed sections is a small disc with radius \( \frac{r}{4} \) and their distance from the axis \( A \) is \( \frac{3}{4} r \).
Mass of each small disc \( \left( m \right) \) is \( \frac{M}{16} \) since the area is \( \frac{1}{16} \) of the original.
The moment of inertia of each small disc about its own center is:
\( I_{\text{small}} = \frac{1}{2} m \left( \frac{r}{4} \right)^2 = \frac{1}{2} \times \frac{M}{16} \times \frac{r^2}{16} = \frac{Mr^2}{512} \)
Using the parallel axis theorem, the moment of inertia about the center of the original disc is:
\( I_{\text{shift}} = I_{\text{small}} + m \left( \frac{3r}{4} \right)^2 \)
\( I_{\text{shift}} = \frac{Mr^2}{512} + \frac{M}{16} \times \frac{9r^2}{16} = \frac{Mr^2}{512} + \frac{9Mr^2}{256} = \frac{19Mr^2}{512} \)
Total for two disks:
\( 2 \times \frac{19Mr^2}{512} = \frac{19Mr^2}{256} \)
Step 3: Moment of Inertia of the Remainder
The remaining moment of inertia is calculated by subtracting the inertia of the removed parts:
\( I_{\text{remainder}} = I_{\text{disc}} - \text{Total } I_{\text{shift}} \)
\( I_{\text{remainder}} = \frac{1}{2} Mr^2 - \frac{19Mr^2}{256} \)
\( = \frac{128Mr^2}{256} - \frac{19Mr^2}{256} = \frac{109Mr^2}{256} \)
Conclusion
The value of \( x \) is 109, confirming it falls within the given range (109,109).
Similar Questions
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
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- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
A block of mass M is attached to a wire. The upper part of wire is made of steel and lower part of wire of copper (as shown in figure). Both parts have the same area of cross section. Neglecting the mass of wires, the ratio of longitudinal strain developed in steel and copper wires is :
(Given, Young's modulus of steel = $2 \ \times \ 10^{11} \ N/m^2$ and Young's modulus of copper = $1.5 \ \times \ 10^{11} \ N/m^2$)
A fly wheel of mass $10$ kg and radius $50$ cm is rotating at a rate of$360$ rpm. Assuming mass to be concentrated at the rim, the constant retarding torque required to stop the rotation of wheel in $6$ rotations will be x Nm. The value of x is ____________.
A body moving horizontally on a smooth surface with a speed of $20$ $ms^{-1}$ splits into two parts and continue to move in the same direction. The masses of two parts are in the ratio of $1:2$. The smaller part moves with a speed of $30$ $ms^{-1}$. The fractional change is the kinetic energy is given by $\frac{1}{x}$. The value of x is__________
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A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2v and v respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :
As shown in the figure, forces of $10^5$ N each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm, shifting the upper face parallel to itself by 0.5 cm. If the side of another cube of the same material is 20 cm, then under similar conditions as above, the displacement will be :
Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
- A body of density '$\rho$' is dropped slowly on the surface of a lake of depth $d$. If the density of the lake water be '$\rho'$' ($\rho' < \rho$) then the time taken by the body to reach the bottom of the lake is
