Problem 37 - Master Resource Book in JEE Main Physics

Question

Two cars of masses \(m_{1}\) and \(m_{2}\) are moving in circles of radii \(r_{1}\) and \(r_{2}\) respectively. Their speeds are such that they make complete circles in the same time \(t\). The ratio of their centripetal acceleration is [AIEEE 2012] (a) \(m_{1} r_{1}: m_{2} 5\) (b) \(m_{1}: m_{2}\) (c) \(r_{1}: r_{2}\) (d) \(1: 1\)

Step-by-Step Solution

Answer

The ratio of centripetal accelerations is \(r_1:r_2\) (Option c).

Step 1: Understand the Problem

We need to find the ratio of the centripetal accelerations of two cars moving in circles. The key information given includes the masses \(m_1\) and \(m_2\), radii \(r_1\) and \(r_2\), and the fact that both cars complete their circles in the same time \(t\).

Step 2: Express Relevant Equations

Centripetal acceleration \(a_c\) is given by the formula \(a_c = \frac{v^2}{r}\), where \(v\) is the linear speed and \(r\) is the radius of the circle. Also, the speed can be expressed in terms of time as \(v = \frac{2\pi r}{t}\).

Step 3: Calculate Speed for Both Cars

Using \(v = \frac{2\pi r}{t}\), calculate \(v_1 = \frac{2\pi r_1}{t}\) for the first car and \(v_2 = \frac{2\pi r_2}{t}\) for the second car.

Step 4: Find Centripetal Acceleration for Each Car

For car 1, the centripetal acceleration is \(a_{c1} = \frac{v_1^2}{r_1} = \frac{\left( \frac{2\pi r_1}{t} \right)^2}{r_1}\). Similarly, for car 2, \(a_{c2} = \frac{v_2^2}{r_2} = \frac{\left( \frac{2\pi r_2}{t} \right)^2}{r_2}\).

Step 5: Simplify the Expressions for Acceleration

Simplify the expressions: \(a_{c1} = \frac{4\pi^2 r_1}{t^2}\) and \(a_{c2} = \frac{4\pi^2 r_2}{t^2}\).

Step 6: Determine the Ratio of Centripetal Accelerations

The ratio \(\frac{a_{c1}}{a_{c2}} = \frac{\frac{4\pi^2 r_1}{t^2}}{\frac{4\pi^2 r_2}{t^2}} = \frac{r_1}{r_2}\).

Step 7: Conclude the Answer

The ratio of the centripetal accelerations is \(r_1:r_2\).

Key Concepts

Understanding Centripetal Acceleration

Centripetal acceleration is a crucial concept when it comes to understanding motion in a circular path. This is the acceleration that keeps an object moving in a circle.
An object moving along a circular path is constantly changing direction, which implies a continuous change in velocity.
Centripetal acceleration is always directed towards the center of the circle.

• The formula for centripetal acceleration is given by: \( a_c = \frac{v^2}{r} \), where \( v \) is the linear speed, and \( r \) is the radius of the circular path.
• This tells us that for a given speed, the smaller the radius of the circle, the greater the centripetal acceleration. Similarly, for a given radius, a higher speed results in higher centripetal acceleration.

Understanding this relationship helps us analyze problems involving circular motion, such as those of cars moving in circular tracks.

Calculating Linear Speed

Linear speed refers to how fast an object is traveling along a circular path. In our case, the cars make a complete circle in a given time, so we can calculate their linear speed.When an object moves in a circle, its linear speed can be found from the circumference of the circle divided by the time taken to complete one full circle.

• Mathematically, it's given by \( v = \frac{2\pi r}{t} \), where \( r \) is the radius and \( t \) is the time for one revolution.
• This equation highlights that the linear speed depends directly on the radius of the path and inversely on the time taken to complete one revolution.
• If two objects move in different circles with different radii but complete their revolutions in the same time, their linear speeds will be different.

By understanding this formula, you can determine how different factors like radius affect the speed of an object in circular motion.

Role of Radius in Circular Path

The radius of a circular path is a key factor that influences the dynamics of circular motion. It's the distance from the center of the circle to any point on its circumference.When dealing with circular motion:

• The radius affects both the centripetal acceleration and the linear speed.
• Specifically, a larger radius results in a longer path, affecting the time it takes to complete a circle if the speed is constant.
• As seen from the formula \( a_c = \frac{v^2}{r} \), when speed is constant, a smaller radius increases the centripetal acceleration.

In problems where you need to find ratios of quantities like accelerations, the radius plays a fundamental role. Understanding the role of radius helps in predicting how changes in it affect motion in a circle.