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 at which the calculator will start sliding, we need to consider the force components acting on it. The force components that come into play are:
1. The gravitational force acting vertically downward (mg), where m is the mass of the calculator and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The force of static friction (Fs) acting parallel to the incline, opposing the gravitational force.
The maximum static friction force (Fs max) can be calculated using the formula:
Fs max = µs N
Where µs is the coefficient of static friction and N is the normal force acting perpendicular to the incline. The normal force is equal to the vertical component of the gravitational force, which is given by:
N = mg cos(θ)
In equilibrium, the force of static friction is equal to the force component that tries to make the calculator slide, which is the horizontal component of the gravitational force:
Fs max = mg sin(θ)
Now, we can set up the equation with the given information:
µs N = mg sin(θ)
Rearranging the equation, we have:
µs mg cos(θ) = mg sin(θ)
Dividing both sides by mg:
µs cos(θ) = sin(θ)
Dividing both sides by cos(θ):
µs = tan(θ)
To find the angle θ, we can use the inverse tangent (arctan) function:
θ = arctan(µs)
Substituting the given coefficient of static friction (µs = 0.7):
θ = arctan(0.7)
Using a calculator or software, we find that:
θ ≈ 35 degrees
Therefore, the calculator will start sliding when the incline angle (θ) reaches approximately 35 degrees.