-PSINO O-Fcose (2) (b) Write Newton's second law for the horizontal and vertical directions. Take rightward to be the +x- direction and vertically upward to be +y-direction. Write your equations in terms of the normal force N, friction f, mass m, and g. =mv² / r (3) =0 (4) (c) Find an equation relating tan(0) with the speed of the bicycle v, radius r, and acceleration due to gravity, Hint: equate the vector decomposition of F (1) and (2) to your results from part (b) - (3) and (4). To type your equations, you can enter Greek letters by selecting the MathType popup button F net, x = Fnet, y =
-PSINO O-Fcose (2) (b) Write Newton's second law for the horizontal and vertical directions. Take rightward to be the +x- direction and vertically upward to be +y-direction. Write your equations in terms of the normal force N, friction f, mass m, and g. =mv² / r (3) =0 (4) (c) Find an equation relating tan(0) with the speed of the bicycle v, radius r, and acceleration due to gravity, Hint: equate the vector decomposition of F (1) and (2) to your results from part (b) - (3) and (4). To type your equations, you can enter Greek letters by selecting the MathType popup button F net, x = Fnet, y =
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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B. for the first part the options are N,f,mg
for the second part, the options are N, N-mg, mg, N-mg-f
![**Physics Problem: Newton’s Second Law**
1. Determine the friction force \( f \):
- o \( F \)
- o \( F\sin\theta \)
- o \( F\cos\theta \)
- o \(-F\sin\theta \)
- o \(-F\cos\theta \)
2. **Writing Newton's Second Law**
(b) Analyze the horizontal and vertical directions. Rightward is the \( +x \)-direction and upward is the \( +y \)-direction. Use normal force \( N \), friction \( f \), mass \( m \), and gravity \( g \):
\[
F_{\text{net},x} = \underline{\quad} = \frac{mv^2}{r} \quad (3)
\]
\[
F_{\text{net},y} = \underline{\quad} = 0 \quad (4)
\]
3. **Finding the Relationship Between Variables**
(c) Derive an equation for \( \tan(\theta) \) using the bicycle’s speed \( v \), radius \( r \), and gravitational acceleration \( g \). Equate vector decomposition of \( F \) from (1) and (2) to results from (b) - (3) and (4):
\[
\tan(\theta) = \sqrt{r}
\]
**Instructions for Typing Equations:**
- Use Greek letters via MathType popup (red radical).
- For Greek letters, select the right-facing arrow near the alpha (\( \alpha \)) symbol.
- Use lowercase for radius \( r \) and velocity \( v \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a3df371-d475-482a-84ce-057c931f1fc6%2Fee86b639-6c2d-4e29-9e39-0b66af4969b0%2F06j8o2_processed.png&w=3840&q=75)
Transcribed Image Text:**Physics Problem: Newton’s Second Law**
1. Determine the friction force \( f \):
- o \( F \)
- o \( F\sin\theta \)
- o \( F\cos\theta \)
- o \(-F\sin\theta \)
- o \(-F\cos\theta \)
2. **Writing Newton's Second Law**
(b) Analyze the horizontal and vertical directions. Rightward is the \( +x \)-direction and upward is the \( +y \)-direction. Use normal force \( N \), friction \( f \), mass \( m \), and gravity \( g \):
\[
F_{\text{net},x} = \underline{\quad} = \frac{mv^2}{r} \quad (3)
\]
\[
F_{\text{net},y} = \underline{\quad} = 0 \quad (4)
\]
3. **Finding the Relationship Between Variables**
(c) Derive an equation for \( \tan(\theta) \) using the bicycle’s speed \( v \), radius \( r \), and gravitational acceleration \( g \). Equate vector decomposition of \( F \) from (1) and (2) to results from (b) - (3) and (4):
\[
\tan(\theta) = \sqrt{r}
\]
**Instructions for Typing Equations:**
- Use Greek letters via MathType popup (red radical).
- For Greek letters, select the right-facing arrow near the alpha (\( \alpha \)) symbol.
- Use lowercase for radius \( r \) and velocity \( v \).
Transcribed Image Text:### Understanding Bicycle Forces and Stability
When riding a bicycle and making a turn, it's crucial to lean at the correct angle to maintain stability. The accompanying free-body diagram illustrates this concept. Here’s a detailed breakdown:
#### Diagram Explanation
- **Free-body Diagram**: The diagram shows a cyclist leaning into a turn. Key forces at play are marked:
- **F**: Total force exerted by the ground.
- **N**: Normal force, perpendicular to the road.
- **f**: Frictional force, parallel to the road.
- **w**: Weight of the cyclist and bicycle.
- **θ (Theta)**: Angle at which the cyclist leans.
- **CG**: Center of Gravity, where all forces balance.
- The cyclist leans at an angle \(\theta\) so the combined effect of \(N\) and \(f\) passes through their center of gravity \(CG\). This balance is crucial to prevent tipping over.
#### Stability Analysis
To remain stable while turning:
- The ground's force must align with the center of gravity, thus balancing the torques.
- The force \(F\) divides into two components:
- **Friction (\(f\))**: Parallel component, providing necessary centripetal force.
- **Normal force (\(N\))**: Vertical component, counteracting the weight of the cyclist and bicycle.
#### Mathematical Expression
##### (a) Express friction and normal force in terms of \(F\) and \(\theta\).
For the vertical balance, the normal force \(N\) must equal the weight \(w\).
Choose the correct expression for the normal force \(N\):
- \(\circ\) \(F\)
- \(\circ\) \(F \sin \theta\)
- \(\bullet\) \(F \cos \theta\)
- \(\circ\) \(-F \sin \theta\)
- \(\circ\) \(-F \cos \theta\)
The correct answer here is \(F \cos \theta\), showing that the normal force balances the gravitational force when leaned correctly. This supports stable turning and prevents the cyclist from falling.
This explanation enhances understanding of the essential physics behind cycling dynamics, crucial for developing advanced cycling techniques.
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