Problem 42 - College Physics

Question

\(\bullet\) A woman takes her dog Rover for a walk on a leash. To get the little pooch moving forward, she pulls on the leash with a force of 20.0 \(\mathrm{N}\) at an angle of \(37^{\circ}\) above the horizontal. (a) How much force is tending to pull Rover forward? (b) How much force is tending to lift Rover off the ground?

Step-by-Step Solution

Answer

Forward force: 15.972 N, Lifting force: 12.036 N.

Step 1: Understanding Force Components

The force exerted on the leash is at an angle, so it can be split into horizontal and vertical components. The horizontal component is the forward pulling force, and the vertical component is the lifting force. To find these components, we will use trigonometry.

Step 2: Calculating the Forward Force

To calculate the forward force (horizontal component), we need to use the cosine of the angle: \[ F_{x} = F \cdot \cos(\theta) \]where \( F = 20.0 \, \mathrm{N} \) and \( \theta = 37^{\circ} \).So, \[ F_{x} = 20.0 \, \mathrm{N} \cdot \cos(37^{\circ}) \approx 20.0 \, \mathrm{N} \cdot 0.7986 = 15.972 \, \mathrm{N} \]

Step 3: Calculating the Lifting Force

To find the lifting force (vertical component), we use the sine of the angle:\[ F_{y} = F \cdot \sin(\theta) \]where \( F = 20.0 \, \mathrm{N} \) and \( \theta = 37^{\circ} \).So,\[ F_{y} = 20.0 \, \mathrm{N} \cdot \sin(37^{\circ}) \approx 20.0 \, \mathrm{N} \cdot 0.6018 = 12.036 \, \mathrm{N} \]

Step 4: Conclusion

The force tending to pull Rover forward is approximately \( 15.972 \, \mathrm{N} \), and the force tending to lift Rover off the ground is approximately \( 12.036 \, \mathrm{N} \).

Key Concepts

Trigonometry in Physics

Trigonometry plays a vital role in understanding and solving physics problems, especially those involving forces. It helps us decompose a force applied at an angle into its vertical and horizontal components. When a force is applied at an angle, it creates a right triangle where the force vector is the hypotenuse. Using trigonometric functions like sine and cosine, we can find the lengths of the legs of this triangle.

• The cosine function (\( \cos \), of an angle) helps in finding the horizontal component.
• The sine function (\( \sin \), of an angle) helps in finding the vertical component.

This approach enables us to analyze forces separately in different directions, which is essential when assessing situations like the angle at which a leash is held to determine the strength of force on different components.

Force Components

In physics, a single force vector can be understood in terms of its components, typically broken down into horizontal and vertical. The force components are critical in understanding the effects of a force applied at an angle.

• The horizontal component (\( F_x \)) is calculated using the cosine of the angle: \( F_x = F \cdot \cos(\theta) \).
• The vertical component (\( F_y \)) uses the sine of the angle: \( F_y = F \cdot \sin(\theta) \).

This method allows us to focus on individual forces separately for different calculations, making it easier to predict the outcome, as seen in situations like a dog being pulled on a leash.

Newton's Laws

Newton's Laws of Motion are fundamental principles that describe the interaction of forces and motion. Newton's Second Law, in particular, is frequently related to force components. It states that the force on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)).
The resolution of a force into components aids in the application of this law. By identifying the individual forces acting in horizontal and vertical directions, we can better understand and predict the acceleration and movement dynamics.
In the exercise with Rover, understanding the effects of forward and lifting forces helps us apply Newton’s laws to anticipate how these force components affect motion and lifting.

Vectors in Physics

Vectors are quantities having both magnitude and direction. In physics, vectors are used to represent physical quantities like force, velocity, and displacement. They are typically represented graphically by arrows, with the length of the arrow corresponding to the vector's magnitude and the direction representing the vector's direction.
Decomposing vectors is essential to simplify complex problems, as forces can act in different directions simultaneously. By resolving a force vector into horizontal and vertical components, we can individually analyze how these components contribute to motion or any physical situation like walking a dog.
This concept is integral to ensuring accurate calculations and understanding of real-world phenomena, where multiple forces interact within a system.