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EXAMPLE 5.5 A simple accelerometer key hanging in a car. In this example we will use Newton's second law to calculate the acceleration of a You tape one end of a piece of string to the ceiling light of your car and hang a key with mass m on the other end of the string (Figure 5.7a). A protractor taped to the light allows you to measure the angle the string makes with the vertical. Your friend drives the car while you make measurements. When the car has a constant acceleration with magnitude a toward the right, the string hangs at rest (relative to the car), making an angle 3 with the vertical. (a) Derive an expression for the acceleration a in terms of the mass m and the measured angle B. (b) In particular, what is a when ß = 45°? When ß = 0? Ta (a) Low-tech accelerometer A FIGURE 5.7 ta A В Mass m T cos BB T si W (b) Free-body di for the key

EXAMPLE 5.5 A simple accelerometer key hanging in a car. In this example we will use Newton's second law to calculate the acceleration of a You tape one end of a piece of string to the ceiling light of your car and hang a key with mass m on the other end of the string (Figure 5.7a). A protractor taped to the light allows you to measure the angle the string makes with the vertical. Your friend drives the car while you make measurements. When the car has a constant acceleration with magnitude a toward the right, the string hangs at rest (relative to the car), making an angle 3 with the vertical. (a) Derive an expression for the acceleration a in terms of the mass m and the measured angle B. (b) In particular, what is a when ß = 45°? When ß = 0? Ta (a) Low-tech accelerometer A FIGURE 5.7 ta A В Mass m T cos BB T si W (b) Free-body di for the key

College Physics
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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**Example 5.5: A Simple Accelerometer** In this example, we use Newton’s second law to calibrate the accelerometer as a piece of hanging keychain in a car. The method involves observing the key's deflection as the car accelerates. 1. **Setup**: Attach a protractor to your car's ceiling and hang a keychain with a small mass from it. This setup allows you to measure the angle \( \beta \) the key makes with the vertical line during acceleration. 2. **Objective**: Derive an expression for the car's acceleration \( a \) in terms of the mass \( m \) and the angle \( \beta \). **Diagram Analysis (Figure 5.7):** - **(a) Low-tech accelerometer**: The diagram illustrates a car accelerating to the right, causing the keychain to swing backward, forming an angle \( \beta \) with the vertical. - **(b) Free-body diagram for the key**: - Shows the forces acting on the key. - The weight \( w = mg \) acts downward. - The tension \( T \) in the string has components: - Vertical: \( T \cos \beta \) balances the weight. - Horizontal: \( T \sin \beta \) equals the mass times acceleration \( ma \). **Solution Steps**: - Apply Newton's second law in the horizontal direction: \( T \sin \beta = ma \). - In the vertical direction, the tension component equals the weight: \( T \cos \beta = mg \). - From these, derive the formula for acceleration \( a = g \tan \beta \). **Applications**: - Determine the acceleration of the vehicle by measuring the angle \( \beta \) when \( \beta = 45^\circ \) or other angles. This simple setup demonstrates how basic principles of physics help us measure motion accurately using everyday objects.
Transcribed Image Text:**Example 5.5: A Simple Accelerometer** In this example, we use Newton’s second law to calibrate the accelerometer as a piece of hanging keychain in a car. The method involves observing the key's deflection as the car accelerates. 1. **Setup**: Attach a protractor to your car's ceiling and hang a keychain with a small mass from it. This setup allows you to measure the angle \( \beta \) the key makes with the vertical line during acceleration. 2. **Objective**: Derive an expression for the car's acceleration \( a \) in terms of the mass \( m \) and the angle \( \beta \). **Diagram Analysis (Figure 5.7):** - **(a) Low-tech accelerometer**: The diagram illustrates a car accelerating to the right, causing the keychain to swing backward, forming an angle \( \beta \) with the vertical. - **(b) Free-body diagram for the key**: - Shows the forces acting on the key. - The weight \( w = mg \) acts downward. - The tension \( T \) in the string has components: - Vertical: \( T \cos \beta \) balances the weight. - Horizontal: \( T \sin \beta \) equals the mass times acceleration \( ma \). **Solution Steps**: - Apply Newton's second law in the horizontal direction: \( T \sin \beta = ma \). - In the vertical direction, the tension component equals the weight: \( T \cos \beta = mg \). - From these, derive the formula for acceleration \( a = g \tan \beta \). **Applications**: - Determine the acceleration of the vehicle by measuring the angle \( \beta \) when \( \beta = 45^\circ \) or other angles. This simple setup demonstrates how basic principles of physics help us measure motion accurately using everyday objects.
**Practice Problem 5.6** *Page 128* At what angle does the hill slope if the acceleration is \( g/2 \)? **Answer: 30°**
Transcribed Image Text:**Practice Problem 5.6** *Page 128* At what angle does the hill slope if the acceleration is \( g/2 \)? **Answer: 30°**
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