Radius of Curvature

Q. If in a vernier caliper 10 VSD coincides with 8 MSD, then the least count of vernier caliper is (given 1 MSD = 1 mm)

Q. Q. A disc of radius R rolls on a rough horizontal surface. The dis†an ce covered by the point A in one revolution is 2ππ R 2R 8R ππ R

Q. At some instant $\overrightarrow{v}=4\hat{i}-3\hat{j}\text{m/s}$ and $\overrightarrow{a}=2\hat{i}+\hat{j}{\text{m/s}}^2$. Find the radius of curvature at that instant.

1. $12.5\text{m}$
2. $25\text{m}$
3. $15\text{m}$
4. $20\text{m}$

Q.

A person is standing on a truck moving with a constant velocity of $14.7m/s$ on a horizontal road. The man throws a ball $v$ that it returns to the truck after the truck has moved $58.8m$. Find the speed and the angle of projection (a) as seen from the truck, (b) as seen from the road.

1. 
2. 
3. 
4. 

Q. A ball is thrown horizontally from the top of a tower with a speed of $5\text{m/s}$ . What will be the radius of curvature of its trajectory $1\text{s}$ after the ball is thrown? Take $g=10{\text{m/s}}^2$.

1. $\frac{35}{2}\sqrt{10}\text{m}$
   
2. $\frac{25}{2}\sqrt{5}\text{m}$
   
3. $\frac{25}{4}\sqrt{10}\text{m}$
   
4. $\frac{25}{4}\sqrt{5}\text{m}$
   

Q. What is the radius of curvature of the parabola traced by a projectile with initial speed $u$ at an angle of $2\theta$ with the horizontal, at a point where the velocity of the particle makes an angle $\theta$ with the horizontal?

1. $\frac{u^2{\cos }^{2}2\theta }{g{\cos }^3\theta }$
   
2. $\frac{u^2{\cos }^{2}2\theta }{g{\cos }^2\theta }$
   
3. $\frac{u^2{\cos }^{3}2\theta }{g{\cos }^2\theta }$
   
4. $\frac{u^2\cos 2\theta }{g{\cos }^3\theta }$
   

Q. Compare the radius of curvature at different points on the curve shown in the figure.

1. $R_A>R_B>R_C$
2. $R_B>R_C>R_A$
3. $R_C>R_B>R_A$
4. $R_B>R_A>R_C$

Q. A particle is projected at angle $\theta$ with horizontal, with velocity $v_0$ at $t=0$. Find the radius of curvature at $t=0$.

1. $v_0^2/g\cos \theta$
2. $v_0^2\cos \theta /g$
3. $(v_0\cos \theta {)}^2/g$
4. $v_0^2/g$

Q. The speed of two bodies A & B undergoing circular motion is same. If the radius of curvature of A is more than B, the centripetal acceleration of A will be more than B.

1. True
2. False

Q. A car of mass $1000\text{kg}$ moves in a circular path with a constant speed of $12\text{m/s}$. It is turned by ${180}^{\circ }$, while travelling a distance of $628\text{m}$ on the road. The centripetal force acting on the car is

1. $1440\text{N}$
2. $320\text{N}$
3. $720\text{N}$
4. $640\text{N}$

Q. The speed of two bodies A & B undergoing circular motion is same. If the radius of curvature of A is more than B, the centripetal acceleration of A will be more than B.

1. True
2. False

Q. What is the speed of a projectile at a point on its path where it makes an angle of ${45}^{\circ }$ with the horizontal? It is given that the radius of curvature ($R$) of particle at this point is $50\sqrt{2}\text{m}$. Take $g=10{\text{m/s}}^2$.

1. $250\sqrt{5}\text{m/s}$
2. $10\sqrt{2}\text{m/s}$
3. $10\sqrt{5}\text{m/s}$
4. $250\sqrt{2}\text{m/s}$

Q. A ball is thrown horizontally from the top of a tower with a speed of $5\text{m/s}$ . What will be the radius of curvature of its trajectory $1\text{s}$ after the ball is thrown? Take $g=10{\text{m/s}}^2$.

1. $\frac{35}{2}\sqrt{10}\text{m}$
   
2. $\frac{25}{2}\sqrt{5}\text{m}$
   
3. $\frac{25}{4}\sqrt{10}\text{m}$
   
4. $\frac{25}{4}\sqrt{5}\text{m}$
   

Q. An object moves on a circular path; such that, its distance covered is given by the function $S=2t^2$. The ratio of magnitudes of its accelerations at times $t_1=1sec$ and $t_2=2$ sec is 1 : 2. The radius of the circle is (Write until two digits after the decimal point)