A Half Ring

Q.

A circular park of radius $20m$ has a road $5m$ wide running around it$.$ Find the cost of metalling the road at $₹14$ per square meter$.\left(Take\mathrm{\pi }=\frac{22}{7}\right)$

Q. Two semicircular rings of linear mass densities $\lambda$ and $2\lambda$ and of radius $3\pi \text{cm}$ each, are joined together to form a complete ring as shown in the figure given below.


Then, distance of the center of the mass of the complete ring from its centre will be:

1. $\frac{9}{8}\text{cm}$
2. $\frac{4}{7}\text{cm}$
3. $3\text{cm}$
4. $2\text{cm}$

Q. The centre of mass of a semi-circular ring lies $5\text{cm}$ above the centre of the semi-circle. What is the circumference of the ring?

1. $20.67\text{cm}$
2. $27.48\text{cm}$
3. $24.67\text{cm}$
4. $18.56\text{cm}$

Q. A semi circular ring of radius $2\pi$ is situated in $X-Y$ plane having centre at $(10,10)$ as shown in the figure below. Find the co-ordinates of centre of mass of the ring.

1. $(12,10)$
2. $(10,14)$
3. $(14,14)$
4. $(10,10)$

Q.

An ideal fluid flows through a pipe of circular cross-section made of two sections with diameters $2.5$cm and $3.75$ cm. The ratio of the velocities in the two pipes is according to the equation of continuity

Q. A cavity of radius b is made in a disc of mass M, radius R, as shown in Fig. Find the new COM :

1. $\frac{-b^2}{R+b}$
2. $\frac{-b^2}{R-b}$
3. $\frac{-R}{2R+b}$
4. $\frac{-R}{3R+b}$

Q. A circular plate of uniform thickness has a diameter $56$cm. A circular portion of diameter $42$cm is removed from one edge as shown in the figure. The centre of mass of the remaining portion from the centre of plate will be.

1. $5$cm
2. $7$cm
3. $9$cm
4. $11$cm

Q. A semi circular ring of radius $2\pi$ is situated in $X-Y$ plane having centre at $(10,10)$ as shown in the figure below. Find the co-ordinates of centre of mass of the ring.

1. $(12,10)$
2. $(10,14)$
3. $(14,14)$
4. $(10,10)$

Q. The centre of mass of a semi-circular ring lies $5\text{cm}$ above the centre of the semi-circle. What is the circumference of the ring?

1. $20.67\text{cm}$
2. $27.48\text{cm}$
3. $24.67\text{cm}$
4. $18.56\text{cm}$

Q. Two semicircular rings of linear mass densities $\lambda$ and $2\lambda$ and of radius $3\pi \text{cm}$ each, are joined together to form a complete ring as shown in the figure given below.


Then, distance of the center of the mass of the complete ring from its centre will be:

1. $\frac{9}{8}\text{cm}$
2. $2\text{cm}$
3. $\frac{4}{7}\text{cm}$
4. $3\text{cm}$

Q. In a vertical U-tube a column of mercury oscillates simple harmonically. If the tube contains $1$ kg of mercury and $1$ cm of mercury column weighs $20$ g, then the period of oscillation is

1. $2$ s
2. $1$ s
3. $\sqrt{2}$ s
4. Insufficient data

Q. A circular wire frame is made of two uniform semicircular wires of same radius and of mass per unit lengths $\lambda$ and $2\lambda$ as shown in figure. Which of the following options is/are true?

1. $x_{cm}=0$
2. $x_{cm}<0$
3. $y_{cm}<0$
4. $y_{cm}=0$