Minimum angle to prevent sliding (Friction & Forces)

AI Thread Summary
To determine the minimum angle of a hill that prevents a refrigerator from slipping during braking, the problem involves balancing forces using Newton's second law. The maximum acceleration while braking is 9 m/s², and the static friction coefficient is 0.4 with the refrigerator mass at 150 kg. The equations derived include the force of friction and gravitational forces, leading to a simplified equation involving trigonometric functions. A suggestion was made to manipulate the equation into a specific trigonometric form to facilitate solving for the angle. The discussion emphasizes the need for trigonometric manipulation rather than calculus to find the minimum angle.
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Homework Statement



Suppose a truck is going up a hill and the maximum acceleration while braking is 9 m/s2. Find the minimum angle of the hill such that the driver can brake at maximum rate without the refrigerator on back slipping. (coefficient of static friction between truck and refrigerator is 0.4 Mass of refrigerator is 150kg)

Homework Equations



Im using Newtons second law and got it down to (force of friction) 588 * cos θ + (force of gravity down the slope) 1470N * sin θ = 1350N (force required to prevent refrigerator from slipping)The above formula can be simplified into purely accelerations :
3.92 * cos θ + 9.8 * sin θ = 9

The Attempt at a Solution



I got it about as far as I mentioned in 2. I'm not sure whether I have to use some calculus to solve this but don't know how I'd go about it because I want the minimum angle. This is when the maximum force of static friction and the force down the slope of gravity = force required to prevent it from slipping.
 
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I agree with your equation. You just need a bit of trig to solve it.
Can you massage it into the form cos(α) sin(θ) + sin(α) cos(θ) = c?
 

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