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CENTRE OF MASS & MOMENT OF INERTIA – MCQ – LEVEL – 1

CENTRE OF MASS & MOMENT OF INERTIA – MCQ – LEVEL – 1

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Question 1 of 10

1. Question
Let $x^2 + y^2 + z^2 = 1$ be the volume bounded below by the cone $x^2 + y^2 = z^2$ and above by the sphere, lying on the positive side of y–axis, also given that the density of the solid will be $\rho( x, y, z) = 1$ . Then consider the following statements :

P : The centre of gravity of the solid will be given by $\left (\frac{\iiint xz \ dxdydz}{V},\frac{\iiint yz \ dxdydz}{V},\frac{\iiint z^2 \ dxdydz}{V} \right )$

Q : The moment of inertia of the solid about z–axis is $\dfrac{\pi}{96}$

Then :
- P is correct but Q is not.
- Q is correct but P is not.
- P & Q both are correct.
- Neither P nor Q is correct.
Correct

Incorrect
Question 2 of 10

2. Question
Consider the following statements about the centre of gravity of the region bounded above by the parabola $y = 4 - x^2$ and below by the x–axis having density $\rho = 2x^2$ .

P : The centre of gravity of the given region will lie on the y–axis.

Q : The x-coordinate of the centre of gravity is equal to $\dfrac{5}{7}$ .

Then which of the above statement is true?
- P
- Q
- Both P and Q
- None of these
Correct

Incorrect
Question 3 of 10

3. Question
The centre of mass for the thin plate bounded by the curves $g(x) = x/2$ and $f(x) = \sqrt{x}$ , $\ 0 \leq x \leq 1$ with density function $\rho = x^2$ is :
- $\left ( \dfrac{192}{601}, \dfrac{7}{601} \right )$
- $\left ( \dfrac{308}{405}, \dfrac{252}{405} \right )$
- $\left ( \dfrac{91}{502}, \dfrac{71}{502} \right )$
- $\left ( \dfrac{111}{305}, \dfrac{113}{305} \right )$
Correct

Incorrect
Question 4 of 10

4. Question
The centre of mass of a thin wire modelled with a semi-circle of radius $a$ with constant density will be :
- $\left ( 0, \dfrac{2a}{\pi} \right )$
- $\left ( 0, \dfrac{4a}{3\pi} \right )$
- $\left ( \dfrac{2a}{\pi}, 0 \right )$
- $\left ( \dfrac{3a}{\pi}, \dfrac{2a}{\pi} \right )$
Correct

Incorrect
Question 5 of 10

5. Question
The centre of mass of the solid lying inside the sphere $x^2 + y^2 + z^2 = b^2$ & outside the sphere $x^2 + y^2 + z^2 = a^2$ with $0 < b < a$ will be :
- $\left ( \pi, 0, 0 \right )$
- $\left ( 0, 2\pi, 0 \right )$
- $\left ( 0, 0, 3\pi \right )$
- $\left ( 0, 0, 0 \right )$
Correct

Incorrect
Question 6 of 10

6. Question
The coordinates of centre of gravity of the area of the region bounded by the curve $x^2 = 9y \ ,$ $y=0$ and $x=6$ will be given by :
- (3.6, 1.2)
- (4.5, 4.5)
- (0, 1.2)
- (4.5, 1.2)
Correct

Incorrect
Question 7 of 10

7. Question
Consider the centre of mass of the tetrahedron in the first octant, bounded by the plane $x + y + z = 1$ . If the density at a point is equal to the sum of the point coordinates then the x–coordinate of the considered centre of mass would be :
- $\dfrac{1}{16}$
- $\dfrac{8}{15}$
- $\dfrac{4}{15}$
- $\dfrac{1}{8}$
Correct

Incorrect
Question 8 of 10

8. Question
The moment of inertia of a right circular cone whose altitude is $</p> <p>h$ and its radius is $r$ about its axis will be given by :
- $\int_{0}^{h}\int_{0}^{y_1}\int_{0}^{x_1}\rho (x^2 + y^2)dxdydz$
- $\int_{0}^{h}\int_{-y_1}^{y_1}\int_{0}^{x_1}\rho (z^2 + y^2)dxdydz$
- $\int_{0}^{h} \int_{0}^{y_{1}} \int_{-x_{1}}^{x_{1}} \rho (y^2 + z^2)dx dy dz$
- $\int_{0}^{h} \int_{-y_{1}}^{y_{1}} \int_{-x^{1}}^{x_{1}} \rho (x^2 + y^2)dxdydz$
Correct

Incorrect
Question 9 of 10

9. Question
The centroid of the area bounded by the parabolas $x = y^2$ and $x^2 = -8y$ will be :
- $\left ( \dfrac{3}{10}, \dfrac{1}{3} \right )$
- $\left ( \dfrac{1}{5}, \dfrac{1}{5} \right )$
- $\left ( \dfrac{9}{5}, \dfrac{-9}{5} \right )$
- $\left ( \dfrac{3}{10}, \dfrac{-1}{3} \right )$
Correct

Incorrect
Question 10 of 10

10. Question
The ordered pair stating the moment of inertia with respect to each coordinate axis of the area between the curve $y = \sin x$ from $x=0$ to $x=\pi$ & the x–axis will be :
- $\left ( \dfrac{2}{9}, \dfrac{\pi^2- 4}{2} \right )$
- $\left ( \dfrac{1}{3}, \dfrac{\pi^2 - 2}{2} \right )$
- $\left ( \dfrac{1}{5}, \dfrac{\pi + 2}{4} \right )$
- $\left ( \dfrac{3}{5}, \dfrac{\pi^2 - 1}{4} \right )$
Correct

Incorrect

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