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?
calculator weight down the hill
mgSinTheta
friction up the hill
mu*mgCosTheta
set them equal, you should get mu=tanTheta when those forces are equal.
Ok thank you.
To determine the angle θ at which the calculator will start sliding, we need to consider the forces acting on the calculator.
1. Gravity: The force due to gravity acts vertically downward. Its magnitude is given by the formula F_g = m * g, where m is the mass of the calculator and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force: The normal force acts perpendicular to the incline and balances the component of gravity acting perpendicular to the incline. Its magnitude is given by the formula F_n = m * g * cos(θ), where θ is the angle of the incline.
3. Friction force: The friction force opposes the motion of the calculator. Its magnitude is given by the formula F_f = μ * F_n, where μ is the coefficient of static friction and F_n is the normal force.
When the calculator is at the point of sliding, the friction force reaches its maximum value, which is equal to the product of the coefficient of static friction and the normal force (i.e., F_f = μ * F_n).
At the point of sliding, the friction force (F_f) is equal to the component of gravity acting parallel to the incline, which is given by F_parallel = m * g * sin(θ).
Therefore, we can set up the equation:
μ * F_n = m * g * sin(θ)
Substituting the equation for the normal force and rearranging, we get:
μ * m * g * cos(θ) = m * g * sin(θ)
Next, we can cancel out the mass (m) and the acceleration due to gravity (g) from both sides of the equation:
μ * cos(θ) = sin(θ)
Now, isolate θ by dividing by cos(θ):
μ = tan(θ)
Finally, take the inverse tangent (or arctan) of both sides to find the angle θ:
θ = arctan(μ)
Given that the coefficient of static friction is 0.7, we can substitute this value into the equation:
θ = arctan(0.7)
Using a calculator or computer software, you would find that arctan(0.7) is approximately 35.3 degrees.
Therefore, the calculator will start sliding at an angle of approximately 35.3 degrees.