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Concept Questions The following table lists data for the speed and the radius in three examples of uniform circular motion. $$ \begin{array}{lcc} & \text { Radius } & \text { Speed } \\ \hline \text { Example 1 } & 0.50 \mathrm{~m} & 12 \mathrm{~m} / \mathrm{s} \\\ \text { Example } 2 & \text { Infinitely large } & 35 \mathrm{~m} / \mathrm{s} \\\ \text { Example 3 } & 1.8 \mathrm{~m} & 2.3 \mathrm{~m} / \mathrm{s} \end{array} $$ (a) Without doing any calculations, identify the example with the smallest centripetal acceleration. (b) Similarly identify the example with the greatest centripetal acceleration. In each case, justify your answer. Problem Find the value for the centripetal acceleration for each example. Verify that your answers are consistent with your answers to the Concept Questions.

Step by step solution

01

Introduction to Centripetal Acceleration

Centripetal acceleration is given by the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the speed and \( r \) is the radius of the circular motion.

02

Analyzing Example 1

For Example 1, we have a radius \( r = 0.50 \, \text{m} \) and speed \( v = 12 \, \text{m/s} \). Substituting these values into the formula gives: \( a_c = \frac{12^2}{0.50} = \frac{144}{0.50} = 288 \, \text{m/s}^2 \).

03

Analyzing Example 2

For Example 2, the radius is infinitely large, and speed is \( v = 35 \, \text{m/s} \). As the radius approaches infinity, the centripetal acceleration formula \( a_c = \frac{v^2}{r} \) approaches zero because the denominator becomes very large. Thus, \( a_c \approx 0 \, \text{m/s}^2 \).

04

Analyzing Example 3

For Example 3, we have a radius \( r = 1.8 \, \text{m} \) and speed \( v = 2.3 \, \text{m/s} \). Substituting these into the formula: \( a_c = \frac{2.3^2}{1.8} = \frac{5.29}{1.8} \approx 2.94 \, \text{m/s}^2 \).

05

Conclusion: Comparing Acceleration Values

From calculations, Example 2 has the smallest centripetal acceleration \( a_c \approx 0 \, \text{m/s}^2 \), and Example 1 has the largest \( a_c = 288 \, \text{m/s}^2 \). This confirms the initial concept question analysis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Uniform Circular Motion

Uniform circular motion occurs when an object moves in a circle at a constant speed. This doesn't mean the object is moving at a constant velocity, because velocity also depends on direction.

In uniform circular motion, the direction of the object changes continuously, even though its speed remains constant. This change in direction implies that there is an acceleration acting upon the object, called **centripetal acceleration**.

Imagine a car driving around a circular track. While its speed might be steady at 30 km/h, the direction of the car is constantly changing to maintain the circular path.

• The object never speeds up or slows down, maintaining constant speed.
• The direction changes continuously, distinguishing it from linear motion.

This constant change in direction is why centripetal force and acceleration are key concepts in uniform circular motion.

Explaining Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path. It's directed towards the center of the circle around which the object is moving. In mathematical terms, centripetal force (F_c) is given by the formula \( F_c = \frac{mv^2}{r} \), where \( m \) is the object's mass, \( v \) is the speed, and \( r \) is the radius of the circular path.

Think of a ball tied to a string, swung in a circle. The tension in the string acts as the centripetal force, pulling the ball towards the center and preventing it from flying off in a straight line.

• Centripetal force is always perpendicular to the object's velocity.
• It's necessary for maintaining the object's circular path.

Without centripetal force, the object would move off in a straight line due to inertia.

Physics Problem Solving with Centripetal Acceleration

When solving physics problems related to centripetal acceleration, it's helpful to start with the formula \( a_c = \frac{v^2}{r} \). This formula allows you to calculate the acceleration based on the object’s speed and the radius of the circle.

Let’s break on down how it works with some examples:

• **Example 1**: Given a small radius of 0.50 m and a speed of 12 m/s, the centripetal acceleration is found to be \( 288 \, \text{m/s}^2 \). This high acceleration is due to the small radius, making the curve tighter.
• **Example 2**: With an infinitely large radius and speed of 35 m/s, the centripetal acceleration approaches zero. A large radius means a gentler curve, requiring less force to change direction.
• **Example 3**: A radius of 1.8 m with a speed of 2.3 m/s results in an acceleration of approximately \( 2.94 \, \text{m/s}^2 \).

Comparing these examples helps illustrate how different speeds and radii affect centripetal acceleration, guiding you to better understand and solve similar problems efficiently.