Problem 80 - Precalculus with Limits

Question

Work A force of 50 pounds, exerted at an angle of \(25^{\circ}\) with the horizontal, is required to slide a desk across a floor. Determine the work done in sliding the desk 15 feet.

Step-by-Step Solution

Answer

First identify the given information (Force, Distance, and Angle). Then convert the angle from degrees to radians, substitute these values into the formula for work (\(Work = Force \times Distance \times cos(Angle)\)), and finally perform the necessary calculations to find the work done.

Step 1: Identify Given Information

From the exercise, the force required (\(F\)) is given as 50 pounds, the distance the desk is moved (\(d\)) is 15 feet, and the angle at which the force is applied (\(\Theta\)) is \(25^{\circ}\).

Step 2: Convert the angle from degrees to radians

Since trigonometric functions in most calculators use radians, it's important to convert the angle from degrees to radians. This can be achieved by using the conversion factor \(\dfrac{\pi}{180}\). Hence, \(\Theta_{rad} = \Theta_{deg} \times \dfrac{\pi}{180} = 25^{\circ} \times \dfrac{\pi}{180}\).

Step 3: Substitute values into the Work formula

Now that we've converted \(\Theta\) to radians and have the values for \(F\) and \(d\), we can substitute these values into the Work formula to calculate work done. Hence, \(Work = F \times d \times cos(\Theta_{rad})\). Thus, substitute \(F = 50\) pounds, \(d = 15\) feet, and \(\Theta_{rad}\), which we calculated in the previous step, into the formula.

Step 4: Calculate Work

After substituting the known values into the formula, perform the multiplication to obtain the work done. Calculate the cosine of the angle, multiply that with the force and the distance to find the work done.

Key Concepts

Force and Distance Calculation

When studying work in physics, a fundamental concept is the relationship between force and distance. Work is defined as the product of the force exerted on an object and the distance over which it moves in the direction of the force. The formula for work (\( W \) is expressed as:
\[ W = F \times d \]
where \( W \) is the work done, \( F \) represents the force applied, and \( d \) signifies the distance moved by the object. In the case of the exercise provided, a force of 50 pounds is used to slide a desk across a distance of 15 feet.

However, whenever the force is applied at an angle, rather than in a straight line, we need to consider the component of the force in the direction of the movement. This is where trigonometric functions come into play. Specifically, the component of the force parallel to the distance moved is found by multiplying the force by the cosine of the angle at which the force is applied. This alteration to the formula accounts for directional force, and the updated work formula becomes: \[ W = F \times d \times \text{cos}(\Theta) \]
Thus, for a force applied at an angle, as in our exercise, it's essential to include this trigonometric adjustment to accurately calculate the work done.

Angle Conversion between Degrees and Radians

In physics, angles can be measured in degrees or radians. When performing calculations, particularly those involving trigonometric functions, it's crucial to use the correct angle unit. Many scientific calculators and programming languages default to radians for trigonometric calculations, which is why angle conversion from degrees to radians is often necessary.

The conversion factor to go from degrees to radians is \( \dfrac{\pi}{180} \), since \( 180^\circ \) corresponds to \( \pi \) radians. To convert degrees to radians, you multiply the angle in degrees by this factor. So, to convert an angle \( \Theta \) in degrees to radians, the formula is: \[ \Theta_{rad} = \Theta_{deg} \times \dfrac{\pi}{180} \]
For instance, in the exercise problem, the given angle of \( 25^\circ \) is converted to radians by multiplying it by \( \dfrac{\pi}{180} \), which would be essential for calculating the trigonometric component of the force correctly.

Trigonometric Functions in Physics

Trigonometric functions play a vital role in physics, especially when dealing with forces, waves, oscillations, and circular motion. Functions like sine, cosine, and tangent are used to relate angles to side ratios in right triangles, which can translate to resolving vector components or determining the path of motion.

In the context of work calculation, when a force is applied at an angle, the cosine function is used to find the magnitude of the force acting parallel to the displacement. The cosine of the angle gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle, which, in physics problems, translates to the component of the force in the direction of movement. \[ W = F \times d \times \text{cos}(\Theta) \]
In the provided exercise, to find the work done by a 50 pound force at a \( 25^\circ \) angle over 15 feet, one must calculate the cosine of the angle (in radians) and then multiply by the force and distance. Such applications of trigonometric functions are ubiquitous in physics, and understanding them is essential to solving problems as well as understanding the physical phenomena they describe.