Question

Once the box begins to slide and accelerate, kinetic friction acts to slow the box with a coefficient of kinetic friction \(\mu_{k}=0.1 .\) The raised end of the surface can be lowered to a point where the box continues sliding but does not accelerate. The critical angle at which this happens, \(\theta_{c}^{\prime},\) can be found from the equation \(\tan \theta_{c}^{\prime}=\mu_{k}\) (a) What is this critical angle for the box? (b) If the box is \(5 \mathrm{ft}\) from the pivot point, at what height will the box stop accelerating?

Short answer

The critical angle is about 5.71° and the height at which the box stops accelerating is approximately 0.497 feet.

Step-by-step solution

Understand the given information

A box is sliding on a surface with a coefficient of kinetic friction \(\mu_{k} = 0.1\). The box continues to slide but does not accelerate when the surface is at a certain critical angle \(\theta_{c}'\). We are to find this critical angle and the height at which the box stops accelerating when it is 5 ft away from the pivot point.

Use the equation for critical angle

The critical angle \(\theta_{c}'\) is given by the equation \[ \tan \theta_{c}' = \mu_{k} \]. We can solve for \theta_{c}'\ using \[ \theta_{c}' = \tan^{-1}(\mu_{k}) \].

Calculate the critical angle

Substitute the given value of \(\mu_{k} = 0.1\) into the equation: \[ \theta_{c}' = \tan^{-1}(0.1) \]. Using a calculator, find the arctan of 0.1: \[ \theta_{c}' \approx 5.71^{\circ} \].

Determine the height

The height \(h\) at which the box stops accelerating is calculated using the relationship between the critical angle and the distance from the pivot point. The relationship is given by \[ h = L \sin \theta_{c}' \], where \(L = 5 \text{ ft}\). Substitute the values: \[ h = 5 \text{ ft} \times \sin(5.71^{\circ}) \]. Using a calculator, find the sine of \(5.71^{\circ}\) and calculate \(h\): \[ h \approx 5 \times 0.0994 = 0.497 \text{ ft} \].

Key concepts

Coefficient of Friction

The coefficient of friction is a measure of how much frictional force exists between two surfaces. In the context of our exercise, the coefficient of kinetic friction, denoted as \( \mu_{k} = 0.1 \), describes the frictional resistance encountered by the sliding box.
This coefficient can vary based on the surface materials involved. For instance, rubber on concrete has a higher coefficient than ice on metal.
It is important to note that the coefficient of kinetic friction is always a positive value and typically less than 1.
This concept is crucial as it directly influences the critical angle at which the box transitions from accelerating to sliding at a constant speed.

Critical Angle

The critical angle \( \theta_{c}' \) is the angle at which the box, while still sliding, stops accelerating and starts moving at a constant speed. This angle is pivotal for understanding the transition between different states of motion.
The critical angle can be found using the equation \[ \tan \theta_{c}' = \mu_{k} \], where \( \mu_{k} \) is the coefficient of kinetic friction.
By rearranging this equation, we get \[ \theta_{c}' = \tan^{-1}(\mu_{k}) \].
Substituting \( \mu_{k} \) with 0.1, we calculate \( \theta_{c}' \) to be approximately 5.71 degrees. This small angle highlights the sensitivity of motion transitions to surface inclinations.

Trigonometric Functions

Trigonometric functions play a significant role in solving problems involving angles and distances. In our exercise, we use the tangent and sine functions.
The tangent function, denoted as \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \], helps find the critical angle \( \theta_{c}' \).
Once we have the critical angle, we use the sine function, \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \], to determine the height at which the box stops accelerating.
In our case, the height \( h \) is computed using \[ h = L \sin \theta_{c}' \], where \( L \) is the distance from the pivot.
For a distance of 5 ft and a critical angle of 5.71 degrees, we find that \( \sin 5.71^{\circ} \approx 0.0994 \). Thus, \[ h \approx 5 \times 0.0994 = 0.497 \text{ ft} \].

Sliding Motion

Sliding motion involves an object moving over a surface while experiencing friction. At the critical angle, the box transitions from accelerating to moving at a constant velocity.
This transition occurs when the gravitational component pulling the object down the incline equals the frictional force resisting its motion.
With \( \theta_{c}'\theta_{c}'= 5.71 \) degrees, the gravitational force down the slope and the frictional resistance balance out.
In practical terms, understanding this concept helps in designing systems where controlled sliding is necessary, ensuring objects can move safely without unintended acceleration.