A 21 kg object is placed on an incline with friction. The incline is raised slowly from zero degrees and at 60.71 degrees, the object begins to slide down the incline. What is the coefficient of static friction? Be as accurate as possible with your answer and be sure to enter at least two decimal places.
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To find the coefficient of static friction, we need to use the angle at which the object begins to slide down the incline. Let's go through the steps to solve this problem.
Step 1: Understand the problem.
We have an object with a mass of 21 kg placed on an incline. The incline is gradually raised, and when it reaches an angle of 60.71 degrees, the object starts sliding down. We need to find the coefficient of static friction.
Step 2: Free body diagram.
Draw a free body diagram to analyze the forces acting on the object. The forces we need to consider are the gravitational force (mg), the normal force (n), and the force of static friction (fs).
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Ff \
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mg
Step 3: Analyze forces along the incline.
We will resolve the forces along the incline to make the calculations easier. The force of gravity can be split into two components: perpendicular (mgcosθ) and parallel (mgsinθ) to the incline.
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Ff \
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mgcosθ
The force of static friction acts upward along the incline to oppose the object from sliding. Its magnitude can be calculated using the equation:
fs = μs * n,
where μs is the coefficient of static friction, and n is the normal force.
Step 4: Calculate the normal force.
Since we have an inclined plane, the normal force is not equal to the object's weight. Instead, it can be calculated using:
n = mgcosθ.
Substituting the known values:
n = (21 kg) * (9.8 m/s^2) * cos(60.71°).
Step 5: Calculate the force of static friction.
Now we can calculate the force of static friction using the equation mentioned earlier:
fs = μs * n.
Substituting the known values:
fs = μs * (21 kg) * (9.8 m/s^2) * cos(60.71°).
Step 6: Equate the force of static friction to zero.
At the point where the object begins to slide, the force of static friction reaches its maximum value before it is overcome, which means the object is at the verge of sliding. Therefore, we can set the force of static friction equal to zero:
fs = μs * (21 kg) * (9.8 m/s^2) * cos(60.71°) = 0.
Solving this equation will give us the coefficient of static friction (μs).
Step 7: Solve for the coefficient of static friction.
To solve for μs, we rearrange the equation and isolate it:
μs = 0 / ((21 kg) * (9.8 m/s^2) * cos(60.71°)).
Calculating this expression will give us the coefficient of static friction.
Remember to enter the answer with at least two decimal places for accuracy.