1. Define angular displacement and angular velocity with formula and SI unit. 2. What is centripetal force? Give one example. 3. What is centrifugal force? Give one example. 4. Obt... 1. Define angular displacement and angular velocity with formula and SI unit. 2. What is centripetal force? Give one example. 3. What is centrifugal force? Give one example. 4. Obtain an expression for the maximum speed with which the cyclist can move without skidding. 5. What is meant by 'banking of road'? 6. A metal bob of mass 0.6 kg is attached to the end of a string of length 1m and the bob is whirled in a horizontal circle with a uniform speed of 12m/s. What is the centripetal force acting on the bob? If the speed of the bob is 5m/s, calculate the tension in the string. 7. What is the maximum speed with which a cyclist can turn around curved path of radius 10.0m, if the coefficient of friction between the tyre and the road is 0.5? If the cyclist wants to avoid overturning by what angle the cyclist must lean from the vertical? 8. A cyclist riding at a speed of 24.25 m/s takes a round a circular road of radius 34.64m. what is his inclination to the vertical? 9. An aeroplane travelling at a speed of 500km/h tilts at any angle of 30° as it makes a turn what is the radius of the curve? 10. What is the angular velocity of: a) a second hand b) hour hand of a clock? 11. The angular displacement of a body is given by θ = 2t² + 5t - 3. Find the value of the angular velocity and angular acceleration when t = 2sec. 12. A stone is tied at the end of a string 1m long and whirled round by holding the other end. The stone makes 120 revolutions in 12 seconds. Calculate angular velocity and linear velocity. 13. An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. what is radius of the loop? 14. A car is moving in a circular horizontal track of a radius 10m with a constant speed of 10m/s. A bob is suspended from the roof of the car by a light rigid rod of length 1m. What is the angle made by the rod with the track? (take g = 10m/s²) Read more

Steps to Solve

1. Define Angular Displacement and Angular Velocity

   • Angular displacement ($\theta$) is the angle through which an object moves on a circular path. It is measured in radians (rad).

   • The formula for angular displacement is: $$ \theta = \frac{s}{r} $$ where $s$ is the arc length and $r$ is the radius of the circle.

   • Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time. It is measured in radians per second (rad/s).

   • The formula for angular velocity is: $$ \omega = \frac{\Delta\theta}{\Delta t} $$

2. Define Centripetal Force with an Example

   • Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle.
   • The formula for centripetal force ($F_c$) is: $$ F_c = \frac{mv^2}{r} $$ where $m$ is the mass of the object, $v$ is the linear speed, and $r$ is the radius of the circular path.
   • Example: A car making a turn on a circular track experiences centripetal force directed towards the center of the track.

3. Centrifugal Force Explanation

   • Centrifugal force is not an actual force but a perceived force that appears when an object in circular motion feels like it is being pushed outward away from the center.
   • It arises from inertia and is observed in a non-inertial reference frame.

4. Max Speed Without Skidding (Cyclist Problem)

   To find the maximum speed ($v_{max}$) without skidding, use: $$ v_{max} = \sqrt{rg} $$ where $r$ is the radius of the curve and $g$ is the acceleration due to gravity.

5. Banking of Roads

   • Banking of roads refers to the inclination of the road surface at a curve to reduce reliance on friction for vehicles to stay on track.
   • The angle of banking ($\theta$) can be calculated using: $$ \tan(\theta) = \frac{v^2}{rg} $$

6. Centripetal Force Example of Metal Bob

   Given the mass $m = 0.6 \text{ kg}$, length $L = 1 \text{ m}$, speed $v = 12 \text{ m/s}$: Calculate the radius ($r$) if the bob is in circular motion. The centripetal force formula applies: $$ F_c = m \cdot \frac{v^2}{L} $$

7. Calculation for Maximum Cyclist Speed

   For a cyclist turning with radius $r = 10 \text{ m}$ and friction coefficient $\mu = 0.5$, the maximum speed can be calculated by: $$ v_{max} = \sqrt{\mu g r} $$ where $g \approx 10 \text{ m/s}^2$.

8. Cyclist on Circular Path Calculation

   With a speed of $v = 24.25 \text{ m/s}$ and radius $r = 34.64 \text{ m}$: Use: $$ \text{Inclination} = \tan^{-1}\left(\frac{v^2}{rg}\right) $$

9. Aeroplane Radius of Curve Calculation

   Speed $= 500 \text{ km/h}$, convert to m/s then: Using the angle $\theta = 30^\circ$: $$ r = \frac{v^2}{g \cdot \tan(\theta)} $$

10. Angular Velocity of Clock Hands

    • For a second hand, it completes 1 revolution per 60 seconds: $$ \omega_{sec , hand} = \frac{2\pi}{60} \text{ rad/s} $$

    • For the hour hand, it completes 1 revolution in 12 hours: $$ \omega_{hour , hand} = \frac{2\pi}{43200} \text{ rad/s} $$

11. Angular and Linear Velocity Calculation

    Given the function $ \theta = 2t^2 + 5t - 3$, differentiate to find angular velocity and substitute $t = 2$ for evaluation.

12. Calculating Angular Velocity of Stone

    With the stone making 120 revolutions in 12 seconds: $$ \text{Angular Velocity} = \frac{120 \cdot 2\pi}{12} \text{ rad/s} $$

13. Radius of Loop for Aircraft

    Given speed $720 \text{ km/h}$, convert to m/s and use: $$ r = \frac{v^2}{g} $$

14. Calculating Angle of Rod with Track

    For radius $10 \text{ m}$ and gravitational acceleration $g = 10 \text{ m/s}^2$: Find the angle $\theta$ using: $$ \sin(\theta) = \frac{g}{v^2/r} $$