Problem 81 - Precalculus with Limits

Question

Work A tractor pulls a log 800 meters, and the tension in the cable connecting the tractor and log is approximately \(15,691\) newtons. The direction of the force is \(35^{\circ}\) above the horizontal. Approximate the work done in pulling the log.

Step-by-Step Solution

Answer

The approximate work done in pulling the log is \(10,213,902.45 J\).

Step 1: Convert the Angle to Radians

As the angle given is in degrees, it must be first converted into radians as most scientific calculators use radians for trigonometric functions. The formula to convert a degree measurement to radian is \( \text{Radians} = \text{Degrees} * \frac{\pi}{180} \). So, the angle is thus converted as follows: \(35^{\circ} = 35 * \frac{\pi}{180} = 0.611 radians \).

Step 2: Apply the Work Formula

Substitute the given values into the work formula, Work = Force x Distance x cos(θ), to obtain Work = \(15,691 N * 800 m * cos(0.611 radian)\).

Step 3: Calculation

Perform the multiplication and use the cosine function to find the final result. Using a scientific calculator, the result comes to approximately \(10,213,902.45 J\), or Joules, the unit of work in the metric system.

Key Concepts

Work Formula Physics

The concept of work in physics is a measure of energy transfer that occurs when a force acts upon an object, causing it to move. Work is done only when components of a force are parallel to the direction of the object’s displacement. The work formula is a valuable tool in calculating the amount of work done in such cases.

The work formula can be expressed as:
\[ W = F \times d \times \text{cos}(\theta) \]
Where:

• \( W \) is work expressed in Joules (J)
• \( F \) is the force applied in newtons (N)
• \( d \) is the distance the object moves in the direction of the force in meters (m)
• \( \theta \) is the angle between the force and the direction of motion

It's important to ensure that the force and distance are in compatible units to avoid any calculation errors. The angle should be taken into account as it impacts the proportion of the applied force that does work along the distance moved.

Force Distance Angle Calculations

When calculating work when a force is applied at an angle, it's important to consider only the component of the force that acts in the direction of the object's displacement. This is where trigonometry comes into play, specifically the cosine function, as it allows us to calculate the effective component of the force doing the work.

In our tractor and log example, although the total force is 15,691 newtons, not all of this force is contributing to moving the log in the intended direction because of the 35-degree angle. The effective force (Fe) can be calculated as follows:
\[ F_e = F \times \text{cos}(\theta) \]
Where:

• \( F \) is the total applied force
• \( \theta \) is the angle in radians

The product of this effective force and the distance the object is moved gives us the work done.

Conversion Degrees to Radians

Angles can be measured in degrees or radians and the conversion between these two units is vital in physics, especially when working with trigonometric functions which often require the angle in radians. The conversion formula is straightforward and essential for accurately applying the work formula.

The formula to convert degrees to radians is given by:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
Therefore, to convert an angle of 35 degrees to radians for our problem, we would calculate:\[ 35^{\text{\textdegree}} \times \frac{\pi}{180} \text{ which equals } 0.611 \text{ radians} \]
It's crucial to perform this conversion before applying the work formula when the provided angle is in degrees to ensure the trigonometric functions yield the correct result.

Trigonometric Functions

Trigonometric functions are fundamental in physics, especially when forces act at angles to the direction of motion. The cosine function in particular, expresses the ratio of the adjacent side to the hypotenuse of a right-angled triangle and is used in the work formula to calculate the component of force that is effective in doing work.

In our scenario, the cosine of the angle in radians is used to find the effective force. Using a scientific calculator to find the cosine of the converted angle (0.611 radians) and multiplying it by the force and the distance provides the work done. The cosine function is just one of many trigonometric functions, each of which has a specific application and significance in various physics problems, making it a valuable area of mathematics to understand fully.