A uniform board is leaning against a smooth vertical wall. The board is at an angle above the horizontal ground. The coefficient of static friction between the ground and the lower end of the board is 0.750. Find the smallest value for the angle , such that the lower end of the board does not slide along the ground.
That would be zero, of course. Care to rephrase the question?
A pipe 0.42 m long and closed at one end is filled with an unknown gas. The third lowest harmonic frequency for the pipe is 780 Hz. (a) What is the speed of sound in the unknown gas? (b) What is the fundamental frequency for this pipe when it is filled with the unknown gas?
To find the smallest value for the angle θ such that the lower end of the board does not slide along the ground, we need to consider the forces acting on the board.
Let's analyze the forces involved:
1. The weight of the board acts vertically downwards.
2. The normal force exerted by the ground acts perpendicular to the ground.
3. The static friction force acts parallel to the ground.
Since the board is not sliding along the ground, the static friction force must be equal to or greater than the component of the weight along the ground.
The component of the weight along the ground can be calculated using trigonometry. It is given by:
Weight along the ground = Weight * sin(θ)
To determine the maximum static friction force, we can multiply the coefficient of static friction (μ) by the normal force.
Maximum static friction force = μ * Normal force
Here, the normal force is equal to the component of the weight perpendicular to the ground, which is given by:
Normal force = Weight * cos(θ)
For the board not to slide, the static friction force must be greater than or equal to the component of the weight along the ground.
Therefore, we have the inequality:
Maximum static friction force ≥ Weight along the ground
Or,
μ * Normal force ≥ Weight * sin(θ)
Now, we can substitute the values we have and find the smallest value of θ:
μ * (Weight * cos(θ)) ≥ Weight * sin(θ)
μ * cos(θ) ≥ sin(θ)
Divide both sides by cos(θ):
μ ≥ tan(θ)
Now, we can find the smallest value of θ by taking the inverse tangent (arctan) of both sides:
θ ≥ arctan(μ)
Therefore, the smallest value for the angle θ, such that the lower end of the board does not slide along the ground, is given by:
θ ≥ arctan(0.750)
Using a calculator, we find:
θ ≥ approximately 36.87 degrees
So, the smallest value for the angle θ is approximately 36.87 degrees.