Problem 5 - University Physics with Modern Physics

Question

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is \(60.0^{\circ} .\) If dog \(A\) exerts a force of 270 \(\mathrm{N}\) and dog \(B\) exerts a force of 300 \(\mathrm{N}\) , find the magnitude of the resultant force and the angle it makes with dog \(A^{\prime}\) s rope.

Step-by-Step Solution

Answer

Resultant force is 484.7 N; angle with dog's A rope is 32.4°.

Step 1: Understand the Forces

We have two forces acting on a point, one with magnitude 270 N (dog A) and another with magnitude 300 N (dog B). The angle between the forces is \( 60.0^{\circ} \). We aim to find the resultant force vector and its angle with respect to dog A's rope.

Step 2: Use the Law of Cosines

To find the magnitude of the resultant force \( F_R \), we use the law of cosines for the triangle formed by the forces. The formula is: \[ F_R = \sqrt{F_A^2 + F_B^2 + 2F_AF_B\cos\theta} \] where \( F_A = 270 \) N, \( F_B = 300 \) N, and \( \theta = 60.0^{\circ} \).

Step 3: Calculate the Resultant Force Magnitude

Substituting the values into the formula: \[ F_R = \sqrt{270^2 + 300^2 + 2 \cdot 270 \cdot 300 \cdot \cos(60.0^{\circ})} \] Solving this gives: \[ F_R = \sqrt{72900 + 90000 + 162000} = \sqrt{234900} \approx 484.67 \text{ N} \]

Step 4: Use the Law of Sines

To find the angle \( \phi \) that the resultant makes with dog A's rope, we use the law of sines: \[ \frac{\sin(\phi)}{F_B} = \frac{\sin(\theta)}{F_R} \] Hence, \( \phi = \arcsin\left( \frac{F_B \cdot \sin(\theta)}{F_R} \right) \).

Step 5: Calculate the Angle with Dog A's Rope

Substituting known values: \[ \phi = \arcsin\left( \frac{300 \cdot \sin(60.0^{\circ})}{484.67} \right) \] \[ \phi = \arcsin\left( \frac{300 \cdot 0.866}{484.67} \right) \approx \arcsin(0.536) \approx 32.4^{\circ} \]

Step 6: Conclusion: Final Result

The resultant force has a magnitude of approximately 484.7 N, and it makes an angle of approximately \( 32.4^{\circ} \) with dog A's rope.

Key Concepts

Resultant Force

When two or more forces are applied to a point, the resultant force is the single force that has the same effect as the original forces combined. In this scenario with the two dogs, the forces are vectors that combine to produce a single resultant force. Understanding the resultant force allows us to simplify complex interactions into a more manageable form. To visualize this, think of vectors as arrows representing both direction and magnitude. The resultant force is like drawing a new arrow from the start to the end of all connected arrows of initial forces.

Law of Cosines

The law of cosines is a helpful tool in trigonometry that lets us find the magnitude of the resultant vector when we know the magnitudes of two vectors and the angle between them. It's given by the formula: \[ F_R = \sqrt{F_A^2 + F_B^2 + 2F_AF_B\cos(\theta)} \]Where:

• \( F_A \) is the force exerted by dog A (270 N).
• \( F_B \) is the force exerted by dog B (300 N).
• \( \theta \) is the angle between the forces (60.0°).

By plugging these values into the formula, we can determine the magnitude of the resultant force which combines both dogs' pulls.

Law of Sines

Once we've determined the magnitude of the resultant force, we use the law of sines to find the angle that this resultant force vector makes with reference to one of the original force vectors. The law of sines relates the ratio of a side length to the sine of its opposite angle, following the formula:\[ \frac{\sin(\phi)}{F_B} = \frac{\sin(\theta)}{F_R} \]Here:

• \( \phi \) is the angle we want to find between the resultant force vector and dog A's rope.
• \( \sin(\theta) \) is the sine of the given angle (60.0°).

This helps us understand not just the size, but also the directional component of the resultant force.

Force Magnitude

In physics, the magnitude of a force describes how strong the force is. It’s a scalar quantity and in this exercise, the magnitudes were 270 N and 300 N. Calculating the resultant magnitude using both the law of cosines and given numbers, yields a single value (approximately 484.7 N), showing the net effect of both forces coming from specific angles. This single number provides a comprehensive understanding of the total force applied at the post.

Angle Calculation

Calculating the angle between the resultant force and dog A's rope is crucial to fully capturing the dynamics of force interactions. With the sine law, we substitute given magnitudes and calculated resultant force to find \( \phi \):\[ \phi = \arcsin\left( \frac{300 \cdot \sin(60.0^{\circ})}{484.67} \right) \approx 32.4^{\circ} \]This means the force is directed 32.4° away from dog A’s rope. Understanding angles is key in vector addition as it affects how forces influence motion and balance. Calculating angles ensures proper alignment and balance of forces in practical applications.