Question

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60.0\(^\circ\). If Rover exerts a force of 270 N and Fido exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with Rover's rope.

Short answer

The magnitude of the resultant force is approximately 493.87 N, and it makes an angle of about 31.6° with Rover's rope.

Step-by-step solution

Understand the Given Problem

We have two forces acting at an angle to each other. Rover exerts a force of 270 N, and Fido exerts a force of 300 N. The angle between the ropes is 60.0 degrees. We need to find the resultant force vector's magnitude and its angle with Rover's rope.

Use the Formula for Resultant Force

To find the magnitude of the resultant force when two forces act at an angle, we use the formula: \[R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}\]where \(F_1 = 270\, N\), \(F_2 = 300\, N\), and \(\theta = 60.0^\circ\).

Calculate Cosine of the Angle

Calculate \(\cos\theta\):\[\cos 60.0^\circ = 0.5\]

Substitute Values into Resultant Force Formula

Substitute the known values into the formula:\[R = \sqrt{270^2 + 300^2 + 2 \times 270 \times 300 \times 0.5}\]

Perform Calculations

Calculate inside the square root:\[= \sqrt{72900 + 90000 + 81000}\]\[= \sqrt{243900}\]Now take the square root:\[R \approx 493.87\, N\]

Find the Angle with Rover's Rope

Use the formula for the direction of the resultant force:\[\tan\phi = \frac{F_2\sin\theta}{F_1 + F_2\cos\theta}\]where \(\phi\) is the angle between the resultant and Rover's force. Substitute the values:\[\tan\phi = \frac{300 \times 0.866}{270 + 300 \times 0.5}\]

Calculate the Angle

Calculate the tangent:\[= \frac{300 \times \sqrt{3}/2}{270 + 150}\]\[= \frac{259.8}{420}\]Calculate \(\phi\):\[\phi \approx \tan^{-1}(0.6186) \approx 31.6^\circ\]

Key concepts

Resultant Force Calculation

When we're dealing with forces, calculating the resultant force is like finding a single force that could replace several individual forces acting on an object, such as the dogs pulling on the ropes in this case. The trick to finding the magnitude of this resultant force is by using a special formula when the forces act at an angle to each other:\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \]Here's how it works:- Suppose two forces, \(F_1\) and \(F_2\), are acting on an object at an angle \(\theta\).- You square the magnitudes of both forces.- Then sum these squares.- Don't forget to add twice the product of the two forces multiplied by the cosine of the angle between them.- Finally, take the square root of the entire expression.For the dogs, Rover exerts \(270\, N\) and Fido \(300\, N\), with an angle of \(60^\circ\) between them. First calculate \(\cos 60^\circ\), which is \(0.5\). Then substitute all values into the formula to find \(R\). This gives us a resultant force of approximately \(493.87\, N\). This result tells us the combined effort in terms of a singular force magnitude.

Vector Angle Determination

Determining the angle a resultant force makes with one of the original forces is about understanding direction as well as magnitude. In our problem, we figure out the angle between the resultant force and Rover's rope using this formula:\[ \tan\phi = \frac{F_2\sin\theta}{F_1 + F_2\cos\theta} \]To grasp this, here's a breakdown:- \(\phi\) is the angle you're solving for, between the resultant force and Rover's direction.- \(\sin\theta\) and \(\cos\theta\) are trigonometric functions based on the angle \(\theta\).- Put the force due to Fido, \(F_2\), on the numerator after multiplying it by \(\sin\theta\).- On the denominator, combine Rover's force \(F_1\) with Fido's force times the cosine of \(\theta\).- Solve \(\phi\) by applying the inverse tangent function.For our exercise, substituting \(F_1 = 270\, N\), \(F_2 = 300\, N\), and \(\theta = 60^\circ\) gives us an angle \(\phi\approx 31.6^\circ\). Thus, the resultant force is tilted about \(31.6\) degrees from Rover's original direction.

Physics Problem Solving

Physics problems often involve breaking things down into manageable parts. Let's apply that idea to figure out how to solve scenarios like our dog scenario with ropes and forces.Here is a simple approach:

• Start by understanding what you are being asked. For example, are you finding a resultant vector or an angle?
• Identify all the given values, and determine the relationships between them, such as forces and angles.
• Use appropriate formulas. For forces, vector addition formulas are your friend. Understand each variable and where it fits.
• Break down the calculation into smaller steps, like solving for \(\cos\theta\) or \(\sin\theta\), to make your life easier.
• Always double-check your results to see if they make sense in a practical context, like ensuring the angle is possible given the situation.

In the dogs' pulling problem, this means recognizing that we are adding vectors, using trigonometry to solve for angles, and making sure the resultant has direction and magnitude that make sense."