### Problem Statement(a) What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 29 km/h and the \( \mu_s \) between tires and track is 0.32?**Explanation:**This problem addresses the concept of centripetal force and friction in a physics context. The aim is to find the smallest radius of a circular track that a bicyclist can navigate safely given a specific speed and coefficient of static friction. Here’s how to approach the problem:1. **Convert the Speed:** - Convert the given speed from km/h to m/s. \[ \text{Speed (v)} = 29 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{29000}{3600} \approx 8.06 \text{ m/s} \]2. **Apply the Centripetal Force Equation:** - The centripetal force necessary to keep the bicycle in a circular motion is provided by the frictional force. \[ F_{\text{centripetal}} = \frac{mv^2}{r} \] - The maximum frictional force is given by: \[ F_{\text{friction}} = \mu_s N = \mu_s mg \] Here, \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the bicyclist, \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \)), and \( N \) is the normal force, which equals \( mg \) on a flat track.3. **Set the Forces Equal:** - Set the centripetal force equal to the maximum frictional force and solve for the radius \( r \). \[ \frac{mv^2}{r} = \mu_s mg \] - The mass \( m \) cancels out: \[ \frac{v^2}{r} = \mu_s g \] - Rearrange to solve for \( r \): \[ r = \frac

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What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 29 km/h and the µ between tires and track is 0.32?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Centripetal Force

A particle, body, object or certain mass while travelling in a rotational path or semicircular path, experiences an inertia force. If that inertia force pulls the towards the inward of the imaginary circle in the rotation, it is termed as centripetal force.

Question

### Problem Statement

(a) What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 29 km/h and the \( \mu_s \) between tires and track is 0.32?

**Explanation:**

This problem addresses the concept of centripetal force and friction in a physics context. The aim is to find the smallest radius of a circular track that a bicyclist can navigate safely given a specific speed and coefficient of static friction.

Here’s how to approach the problem:

1. **Convert the Speed:**
- Convert the given speed from km/h to m/s.

\[
\text{Speed (v)} = 29 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{29000}{3600} \approx 8.06 \text{ m/s}
\]

2. **Apply the Centripetal Force Equation:**
- The centripetal force necessary to keep the bicycle in a circular motion is provided by the frictional force.

\[
F_{\text{centripetal}} = \frac{mv^2}{r}
\]

- The maximum frictional force is given by:

\[
F_{\text{friction}} = \mu_s N = \mu_s mg
\]

Here, \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the bicyclist, \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \)), and \( N \) is the normal force, which equals \( mg \) on a flat track.

3. **Set the Forces Equal:**
- Set the centripetal force equal to the maximum frictional force and solve for the radius \( r \).

\[
\frac{mv^2}{r} = \mu_s mg
\]

- The mass \( m \) cancels out:

\[
\frac{v^2}{r} = \mu_s g
\]

- Rearrange to solve for \( r \):

\[
r = \frac

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