Problem 84 - Precalculus with Limits

Question

Work A ski patroller pulls a rescue toboggan across a flat snow surface by exerting a force of 35 pounds on a handle that makes an angle of \(22^{\circ}\) with the horizontal. Find the work done in pulling the toboggan 200 feet.

Step-by-Step Solution

Answer

The work done pulling the toboggan 200 feet is equal to \(35*200*cos(\frac{11\pi}{90})\) foot-pounds.

Step 1: Convert an Angle to Radians

Convert the angle from degrees to radians. To do this, use the formula \( rad = deg* \rac{\pi}{180} \), where deg is the number of degrees. This is needed because trigonometric functions in standard mathematics operate on angles expressed in radians. In this case, substitute \(22^{\circ}\) into the formula, yielding \(rad=22*\frac{\pi}{180}=\frac{11\pi}{90}\).

Step 2: Find the Scalar Product of Vectors

Calculate the work done as the scalar product of the force and displacement vectors. The formula for this is \(Work = |Force|*|Displacement|*cos(\theta) \) where |Force| and |Displacement| are the magnitudes of the respective vectors, and \(\theta\) is the angle between them. Substituting the given values, we get: \( Work=35*200*cos(\frac{11\pi}{90}) \).

Step 3: Calculate the Result

Calculate the result of the expression obtained in Step 2. This will give the answer to the problem.

Key Concepts

Angle Conversion

When working with angles in physics problems, especially those involving trigonometric functions, it's common to convert angles from degrees to radians. Why is that? Simply because most mathematical formulas involving trigonometry are based on radian measure.

The conversion is straightforward. The formula is:

• To convert degrees to radians, use
  \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

In this context, given an angle of \(22^{\circ}\), the conversion looks like this:

• \( 22 \times \frac{\pi}{180} = \frac{11\pi}{90} \)

This conversion allows us to use the cosine function accurately in physics calculations.

Knowing how to convert angles will be helpful not just in assignments but in understanding how trigonometry applies in real-world scenarios like this one, involving work and forces.

Scalar Product

The concept of a scalar product, or dot product, is essential in calculating work done by a force. It involves multiplying two vectors to give a scalar value, which in this scenario equates to the work done.

The formula for calculating the scalar product is:

• \[ \text{Work} = |\mathbf{F}| \times |\mathbf{d}| \times \cos(\theta) \]

Where:

• \(|\mathbf{F}|\) is the magnitude of the force vector
• \(|\mathbf{d}|\) is the displacement distance
• \(\theta\) is the angle between the force and displacement directions

In the given problem, the force is 35 pounds, the displacement is 200 feet, and the angle \(\theta\) in radians comes out to be \(\frac{11\pi}{90}\). Therefore, the work done can be calculated by plugging these values into the mentioned formula.

Understanding the scalar product helps you connect physical actions to mathematical calculations, offering a way to quantify the work involved in moving objects.

Force and Displacement

For calculating work, it's crucial to understand the relationship between force and displacement. Work is essentially a product of how much force is exerted along a path and how long that path is.

Picture this: you exert a force on an item, like a sled, and that force has to counteract friction or other resistive forces to move the sled forward. That's where displacement comes in. It measures how far the object moves in the direction of the applied force.

In our scenario, you have:

• A force of 35 pounds exerted on the handle.
• The distance, or displacement, is 200 feet.
• The force is applied at an angle of \(22^{\circ}\) (or \(\frac{11\pi}{90}\) in radians) with the horizontal ground.

Grasping how these components interact allows you to see the complete picture of how work is calculated in physics. It builds a bridge between theoretical concepts and their practical application in everyday activities.