Radius of Curvature

Q. Radius of the curved road on national highway is R. Width of the road is b. The outer edge of the road is raised by hwith respect to inner edge so that a car with velocity vcan pass safe over it. The value of his

1. $\frac{v^{2}b}{Rg}$
2. $\frac{v}{Rgb}$
3. $\frac{v^{2}R}{g}$
4. $\frac{v^{2}b}{R}$

Q. A particle is projected with speed $u$ at an angle $\theta$ with the horizontal. Find the radius of curvature at the highest point of its trajectory.

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

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

Q.

What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle $\frac{\theta }{2}$ with the horizontal ?

Q. A stone of mass $2\text{kg}$ is projected from ground with speed $20\text{m/s}$ at angle ${30}^{\circ }$ with horizontal. The centripetal force acting on the object $1s$ after projection is

1. $10\text{N}$
2. $20\text{N}$
3. $30\text{N}$
4. $40\text{N}$

Q.

If a cyclist moving with a speed of 4.9 m/s on a level road can take a sharp circular turn of radius 4 m, then coefficient of friction between the cycle tyres and road is

1. 0.61

2. 0.71

3. 0.51

4. 0.41

Q. An object is thrown horizontally with a velocity of $10\text{m/s}$. Find the radius of curvature of its trajectory $3\text{s}$ after the motion has begun. Take $g=10{\text{m/s}}^2$

1. $316\text{m}$
2. $100\sqrt{5}\text{m}$
3. $100\sqrt{10}\text{m}$
4. $300\text{m}$

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. $640\text{N}$
4. $720\text{N}$

Q. If the pressure due to liquid varies with the depth $y$ as shown in figure, then the density of the liquid is $(g=10m/s^2)$

1. $0.075kg/m^3$
2. $0.133kg/m^3$
3. $0.75kg/m^3$
4. $1.33kg/m^3$

Q. A particle is moving in the $x-y$ plane with uniform speed $v$. The equation of trajectory for the particle is represented by the parabola $y=a{x}^2,$ where $a$ is a $+ve$ constant. Find the radius of curvature of trajectory at the point $x=0$.

1. $\frac{a}{2}$
2. $\frac{1}{2a}$
3. $\frac{4}{a}$
4. $\frac{3}{2}a$

Q.

A train is moving towards north. At one place it turns towards north-east, here we observe that

1. The radius of curvature of outer rail will be greater than that of the inner rail

2. The radius of the inner rail will be greater than that of the outer rail

3. The radius of curvature of the outer and inner rails will be the same

4. The radius of curvature of one of the rails will be greater

Q. a small square shaped loop of edge length l is placed at the centre of a circular loop a of radius r with its plane perpendicular to plane of the circular loop the coefficient of mutual induc†an ce is

Q. A car of mass $1000\text{kg}$ moves in a circular path with a constant speed of $16\text{m/s}$. It is turned by ${90}^{\circ }$, while travelling a distance of $628\text{m}$ on the road. The centripetal force acting on the car is

1. $160\text{N}$
2. $320\text{N}$
3. $1280\text{N}$
4. $640\text{N}$

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

Q. A stone is fastened to one end of a string and is whirled in a vertical circle of radius $10\text{m}$. Find the minimum speed the stone can have at the highest point on the circle.
[Take$g=10{\text{m/s}}^2$]

1. $10\text{m/s}$
2. $12\text{m/s}$
3. $5\text{m/s}$
4. $0$

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

Q. A particle is moving with velocity $\overrightarrow{v}=k(y\hat{i}+x\hat{j})$ where $k$ is constant. The general equation of the path $(c=constant)$ is

1. $y=x^2+constant$
2. $y^2=x+constant$
3. $xy=constant$
4. $y^2=x^2+constant$

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

Q. A stone of mass $2\text{kg}$ is projected from ground with speed $20\text{m/s}$ at angle ${30}^{\circ }$ with horizontal. The centripetal force acting on the object $1s$ after projection is

1. $10\text{N}$
2. $20\text{N}$
3. $30\text{N}$
4. $40\text{N}$

Q. A shell of mass $m$ is projected with a velocity $v$ at an angle ${60}^o$ to horizontal. When it reaches the maximum height, its angular momentum with respect to point of projection is

1. $\frac{\sqrt{3}{mv}^3}{8g}$
2. $\frac{3{mv}^3}{8g}$
3. $\frac{3{mv}^3}{16g}$
4. $\frac{\sqrt{3}{mv}^3}{16g}$

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

Q. A vehicle can travel along a curved road with a speed of $18km{h}^{-1}$. If the total centripetal force is provided by the banking of the road and the radius of the curvature is $10$ m, calculate the banking angle of the road.

1. ${90}^o$
2. ${30}^o$
3. ${14}^o$
4. ${16}^o$

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

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

Q. A train is moving on a circular curve of radius 1500m at the rate of 66 km/h. Through what angle has it turned in 10 seconds?

Q.

A particle is projected at angle θ with horizontal with velocity $V_0$

Find

(i) tangential and normal acceleration of the particle at t = 0 and at highest point of its trajectory.

(ii) radius of curvature at t = 0 and highest point.

1. at

2. At highest point

3. at

4. At highest point

Q. The velocity gradient at a distance $100mm$ from the boundary if the velocity profile is a parabola with the velocity at $200mm$ from the boundary between the boundary layer and free stream flow be $2m/s$ is $se{c}^{-1}$

1. 
   $1$
2. 
   $1.5$
3. 
   $2$
4. 
   $2.33$

Q. Velocity of a particle moving along positive $x$-direction is $V=(40-10t)m/s$. Here, $t$ is in seconds. At time $t=0$, the $x$ coordinate of particle is zero. The time when the particle is at a distance of $60m$ from origin :

1. 2 s
2. 4 s
3. 6 s
4. 8 s

Q. A particle is projected upwards with a velocity of $100m/sec$ at an angle of ${60}^0$ with the vertical. The time (in sec) when the particle will move perpendicular to its initial direction is 10x. (taking $g=10m/se{c}^2$). The value of x is:

Q. A boy aims at a bird from a point at a horizontal distance of $100m$. The gun can impart a velocity of $500$ $m{s}^{-1}$ to the bullet. At what height above the bird must he aim his gun in order to hit $(g=10$ $m{s}^{-2}$) :

1. $20cm$
2. $50cm$
3. $100cm$
4. $40cm$