You are very slowly pushing up an incline with a calculator sitting on it. The coefficient of static friction is 0.7. At what angle θ will the calculator start sliding?
To determine the angle θ at which the calculator will start sliding, we need to calculate the maximum angle that the incline can have before the force of gravity exceeds the force of static friction.
First, let's consider the forces acting on the calculator on the incline. We have the force of gravity acting downward, which can be decomposed into two components: the perpendicular component (mg*cosθ), and the parallel component (mg*sinθ), where m is the mass of the calculator, and g is the acceleration due to gravity.
The force of static friction acts parallel to the incline and opposes the motion, preventing the calculator from sliding. The equation for static friction is given by fs ≤ μs * Fn, where fs is the force of static friction, μs is the coefficient of static friction, and Fn is the normal force acting perpendicular to the incline.
To find the normal force Fn, we need to consider the perpendicular component of the force of gravity:
Fn = mg*cosθ
Since the calculator is just on the verge of sliding, the force of static friction is at its maximum and equal to the limiting value:
fs = μs * Fn = μs * mg*cosθ
At the point of impending motion, the force of static friction is equal to the parallel component of the force of gravity:
fs = mg*sinθ
Setting these two expressions equal to each other, we have:
μs * mg*cosθ = mg*sinθ
Dividing both sides by mg and canceling out common terms, we get:
μs*cosθ = sinθ
Dividing both sides by cosθ, we have:
μs = tanθ
Now, we can solve for the angle θ by taking the inverse tangent (arctan) of the coefficient of static friction:
θ = arctan(μs)
Substituting the given coefficient of static friction μs = 0.7, we find:
θ ≈ 35.26 degrees
Therefore, the calculator will start sliding when the incline reaches an angle of approximately 35.26 degrees.