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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Question
Practice problem: what ramp angle would be needed in order for the cable tension to equal one-half of the car’s weight?
Answer 30 degrees
Transcribed Image Text:**Example 5.3: Car on a Ramp**
**Solution**
**Set Up:** Figure 5.3b shows our free-body diagram for the car. The three forces exerted on the car are its weight (magnitude \( w \)), the tension \( T \) in the cable (magnitude \( T \)), and the normal force \( \mathbf{N} \) that the track exerts. We orient our coordinate axes parallel and perpendicular to the ramp, and we replace the weight force by its components.
**Solution:** The car is in equilibrium, so we first find the components of each force in that coordinate system and then apply Newton's first law. To find the components of the weight, we note that the angle \( \alpha \) between the vertical and the tension \( \mathbf{T} \) is equal to the angle \( \alpha \) of the incline. The components of the weight are \( w \sin \alpha \) down the ramp and \( w \cos \alpha \) perpendicular to the ramp. The normal force has components \( N \sin \alpha \) up the ramp and \( N \cos \alpha \) perpendicular to it. The applied force \( \mathbf{T} \) has only a component up the ramp. \(\sum F_x = 0\), \(T = w \sin \alpha\). \(\sum F_y = 0\), \(N = w \cos \alpha\).
**Diagrams:**
- **(a) Car on ramp:** This diagram visually shows the car positioned on a sloped ramp pulled by a cable.
- **(b) Free-body diagram for car:** This diagram includes vectors \( w \sin \alpha \) and \( w \cos \alpha \), representing the components of the weight force, with a line indicating these components’ directions.
This setup helps visualize and solve the problem of a car on an inclined plane by breaking down forces into components and applying equilibrium conditions.
![**Example 5.4: Lifting Granite and Dumping Dirt**
Let's examine an inclined plane problem, involving two objects connected by a cable. Blocks of granite, each with weight \( w \), are elevated for environmental reasons, while dirt is dumped into a hole. The system includes a conveyor belt lifting granite and a mechanism for dumping dirt.
**Equation for Tension and Angle:**
- The balance of forces is analyzed using Newton's Laws.
- For equilibrium, the equations used are:
\[
\Sigma F = 0
\]
\[
n + (w \cos \alpha) = 0
\]
\[
T = w \sin \alpha
\]
With:
\[
T = w \sin \alpha
\]
\[
n = w \cos \alpha
\]
To ensure proper comprehension of the setup, remember:
- Vectors should align with the correct coordinate axes.
- By definition, \( T \) and \( n \) are positive.
Upon solving for \( T \) and \( n \), we find:
\[
T = w \sin \alpha
\]
\[
n = w \cos \alpha
\]
Numerical Example:
- Substituting given values \( w = 1.76 \times 10^4 \) N and \( \alpha = 26^\circ \):
\[
n = (1.76 \times 10^4 \text{ N}) \cos 26^\circ = 1.58 \times 10^4 \text{ N}
\]
\[
T = (1.76 \times 10^4 \text{ N}) \sin 26^\circ = 7.72 \times 10^3 \text{ N}
\]
**Reflect:**
The example shows the impact of specific conditions on tension. A horizontal incline at \( \alpha = 0 \) leads to zero cable tension \( (T = 0) \), consistent with expectations.
**Practice Problem:**
Determine the ramp angle needed for the cable tension to equal half the car’s weight. Answer: \( 30^\circ \).
The problem presented demonstrates typical applications of Newton's laws to real-world situations involving inclined planes and tension in cables. Understanding these principles is crucial for solving similar physics problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28764828-4d1c-44b8-9544-7d79c8b69e6b%2F9d72e965-acdf-4d42-9791-49001477588a%2Fgtjzjcn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example 5.4: Lifting Granite and Dumping Dirt**
Let's examine an inclined plane problem, involving two objects connected by a cable. Blocks of granite, each with weight \( w \), are elevated for environmental reasons, while dirt is dumped into a hole. The system includes a conveyor belt lifting granite and a mechanism for dumping dirt.
**Equation for Tension and Angle:**
- The balance of forces is analyzed using Newton's Laws.
- For equilibrium, the equations used are:
\[
\Sigma F = 0
\]
\[
n + (w \cos \alpha) = 0
\]
\[
T = w \sin \alpha
\]
With:
\[
T = w \sin \alpha
\]
\[
n = w \cos \alpha
\]
To ensure proper comprehension of the setup, remember:
- Vectors should align with the correct coordinate axes.
- By definition, \( T \) and \( n \) are positive.
Upon solving for \( T \) and \( n \), we find:
\[
T = w \sin \alpha
\]
\[
n = w \cos \alpha
\]
Numerical Example:
- Substituting given values \( w = 1.76 \times 10^4 \) N and \( \alpha = 26^\circ \):
\[
n = (1.76 \times 10^4 \text{ N}) \cos 26^\circ = 1.58 \times 10^4 \text{ N}
\]
\[
T = (1.76 \times 10^4 \text{ N}) \sin 26^\circ = 7.72 \times 10^3 \text{ N}
\]
**Reflect:**
The example shows the impact of specific conditions on tension. A horizontal incline at \( \alpha = 0 \) leads to zero cable tension \( (T = 0) \), consistent with expectations.
**Practice Problem:**
Determine the ramp angle needed for the cable tension to equal half the car’s weight. Answer: \( 30^\circ \).
The problem presented demonstrates typical applications of Newton's laws to real-world situations involving inclined planes and tension in cables. Understanding these principles is crucial for solving similar physics problems.
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