Problem 39 - Master Resource Book in JEE Main Physics

Question

What is the smallest radius of a circle at which a cyclist can travel if its speed is \(36 \mathrm{kmh}^{-1}\), angle of inclination is \(45^{\circ}\) and \(g=10 \mathrm{~ms}^{-2}\) ? (a) \(20 \mathrm{~m}\) (b) \(10 \mathrm{~m}\) (c) \(30 \mathrm{~m}\) (d) \(40 \mathrm{~m}\)

Step-by-Step Solution

Answer

The smallest radius is 10 m (option b).

Step 1: Convert Speed to Meters per Second

The cyclist's speed is given as \(36 \ \mathrm{kmh^{-1}}\). First, convert this speed to meters per second. Use the conversion factor: \[1 \ \mathrm{kmh^{-1} = \frac{1000}{3600} \ \mathrm{ms^{-1}}}\]So, \[36 \ \mathrm{kmh^{-1}} = 36 \times \frac{1000}{3600} \ \mathrm{ms^{-1}}\ = 10 \ \mathrm{ms^{-1}}\]

Step 2: Recall the Formula for Banking Angle and Centripetal Force

For a cyclist turning in a circular path with angle of inclination \(\theta\), radius \(r\), and speed \(v\), the centripetal force is provided by gravity acting at an angle. The formula linking these is:\[ \tan(\theta) = \frac{v^2}{rg} \]where \(\theta = 45^\circ\), \(v = 10 \ \mathrm{ms^{-1}}\), and \(g = 10 \ \mathrm{ms^{-2}}\).

Step 3: Solve for Radius

Plug the known values into the formula:\[ \tan(45^\circ) = \frac{10^2}{r \times 10} \]Since \(\tan(45^\circ) = 1\), the equation becomes:\[ 1 = \frac{100}{10r} \]\[ r = \frac{100}{10} = 10 \ \mathrm{m} \]

Step 4: Verify the Calculated Answer

Ensure that the calculated radius \(r = 10 \ \mathrm{m}\) matches one of the options provided.The options are:\((a) \ 20 \ \mathrm{m}\), \((b) \ 10 \ \mathrm{m}\), \((c) \ 30 \ \mathrm{m}\), \((d) \ 40 \ \mathrm{m}\).The calculation shows the smallest radius to be \(10 \ \mathrm{m}\), which matches option \((b)\).

Key Concepts

Centripetal Force

Imagine you are a cyclist turning along a circular path. When moving in a circle, a special type of force called "centripetal force" is at work. This force pulls the cyclist towards the center of the circle, enabling the turn without falling off the path.

• This force does not exist by itself but is a result of other forces, such as gravity or friction, acting in the radial direction.
• In this case, as the cyclist leans into the curve, gravity plays a role in helping create the centripetal force.
• For a given speed and radius, the centripetal force needed can be calculated using the formula: \[ F_c = \frac{mv^2}{r} \] where \( F_c \) is the centripetal force, \( m \) is the mass of the cyclist and bicycle, \( v \) is the speed, and \( r \) is the radius.

The formula shows us why you can't take sharp turns at high speeds without risking a fall. The faster you go, the more centripetal force you need!

Banking Angle

Cyclists and drivers often encounter the concept of the "banking angle" when taking turns. This angle refers to the inclination formed when you lean into a curve.

• The purpose of this inclination is to help maintain balance while taking a curve.
• The banking angle reduces the reliance on friction to provide the necessary centripetal force, as it uses components of gravitational force.
• In mathematical terms, the relationship is expressed as: \[ \tan(\theta) = \frac{v^2}{rg} \] where \( \theta \) is the banking angle, \( v \) is the velocity, \( r \) is the radius of the circular path, and \( g \) is the gravitational acceleration.

By tilting inwards, riders intelligently use gravity to help themselves turn, making the turn smoother and reducing the risk of skidding.

Speed Conversion

Before tackling any problem involving speed, it's crucial to ensure that all units match for precision in calculations. Often, questions provide speed in kilometers per hour \((\mathrm{kmh^{-1}})\), but scientific formulas use meters per second \((\mathrm{ms^{-1}})\). To convert speed from \(\mathrm{kmh^{-1}}\) to \(\mathrm{ms^{-1}}\), you can use a simple conversion factor:

• \[ 1 \ \mathrm{kmh^{-1} = \frac{1000}{3600} \ \mathrm{ms^{-1}}} \] This conversion factor originates from the definitions of a kilometer \((1000 \ \mathrm{m})\) and an hour \((3600 \ \mathrm{s})\).
• For example, if a cyclist is traveling at \(36 \ \mathrm{kmh^{-1}}\), you can convert it as follows:\[ 36 \ \mathrm{kmh^{-1}} \ = 36 \times \frac{1000}{3600} \ \mathrm{ms^{-1}} \ = 10 \ \mathrm{ms^{-1}} \]

By converting the speed, you ensure alignment in units when applying these values to formulas, which is essential for correct and accurate calculations.