Question

The coefficient of static friction between Teflon and scrambled eggs is about \(0.04\). What is the smallest angle from the horizontal that will cause the eggs to slide across the bottom of a Teflon-coated skillet?

Short answer

The smallest angle from the horizontal that will cause the eggs to slide across the bottom of a Teflon-coated skillet is approximately \( 2.29 \) degrees.

Step-by-step solution

Understand the Physical Laws Involved

The force of static friction can be calculated using the formula \( F_s = \mu_s * F_n \) where \( F_s \) is the force of static friction, \( \mu_s \) is the coefficient of static friction and \( F_n \) is the normal force. However, in this situation, the normal force isn't given directly. When the eggs start to slide, it means that the downward force caused by gravity along the angle equals the force of static friction. As we only need the angle causing this situation, we can ignore the mass of the eggs - it would cancel out on both sides of the equation. So, we can also write \( F_g * sin(\Theta) = \mu_s * F_g * cos(\Theta) \) where \( F_g \) is the force due to gravity, \( \Theta \) is the angle of inclination, and \( sin \) and \( cos \) are trigonometric functions which values depend on the angle \( \Theta \). For the given situation, \( \mu_s \) is 0.04.

Set Up the Equation and Solve for the Unknown

Rearranging the formula results in \( tan(\Theta) = \mu_s \) as \( sin(\Theta) / cos(\Theta) = tan(\Theta) \). From the problem, we know that the coefficient of friction \( \mu_s \) is 0.04. So we solve for \( \Theta \) as \( tan^{-1}(0.04) \). The inverse tangent function \( tan^{-1} \) gives us the angle whose tangent is the number inside the parentheses.

Translate to Degrees

The result from step 2 will be in radians. However, the problem asks for the angle in degrees. To convert from radians to degrees, multiply the result by \( 180 / \pi \).

Key concepts

Angle of Inclination

An angle of inclination is a measure of how steeply a surface is tilted with respect to the horizontal. In practical situations, this angle determines when an object begins to move due to gravity overcoming friction. Imagine holding a book flat in your hand with a ball resting on it. As you start tilting the book, at a certain angle, the ball will start sliding down. This inclination angle is crucial when dealing with surfaces where friction is involved, such as a skillet in our scenario.

• If the angle of inclination is small, the object's weight component parallel to the surface isn't enough to overcome static friction.
• If the angle is increased, at some point, even a tiny push could set the object in motion.

It's the angle at which static friction just fails to hold the object stationary. Understanding and calculating the angle of inclination helps in designing slopes and angles in engineering and everyday applications.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between side lengths and angles of triangles. It's central to solving problems involving angles and distances, especially when dealing with inclined planes and friction. In our problem, trigonometry helps us relate the angle of inclination to the forces parallel and perpendicular to the inclined surface.
The essence of trigonometry in this context:

• Sine (\( \sin(\Theta) \)): Relates the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (\( \cos(\Theta) \)): Relates the adjacent side to the hypotenuse.
• Tangent (\( \tan(\Theta) \)): A combination, relating the opposite side to the adjacent side, crucial in expressing the ratio of forces on an incline.

In our friction problem, we used the tangent function because it naturally combines sine and cosine to relate the angle to friction forces. Mastery of trigonometric functions simplifies dealing with complex inclined plane problems.

Inverse Tangent

Inverse tangent, denoted as \( \tan^{-1} \) or arctan, is a function that helps us find an angle when we know the tangent value of that angle. In real-life applications like our friction problem, finding the inverse tangent allows us to determine the precise angle required for motion to occur due to gravity overcoming static friction.
Understanding and applying inverse tangent involves:

• Using \( \tan^{-1} \) to find the angle \( \Theta \), where \( \tan(\Theta) \) equals a known value, such as the coefficient of friction.
• Recognizing that the value obtained is in radians and may need to be converted into degrees for practical understanding.
• Recognizing its importance in navigation, physics, and engineering tasks where specific angle measurements are required.

Using the inverse tangent is straightforward with calculators or software tools, and it bridges our understanding between mathematical abstractions and real-world angles.