Final answer:
The maximum angle at which a cart will not move up a ramp is equal to the inverse tangent (arctan) of the coefficient of static friction.
Step-by-step explanation:
To find the angle at which a cart will not move up a ramp, we consider the maximum static frictional force that must equal the component of the cart's weight parallel to the incline. We use the equation of static friction, μs×N, where μs is the static friction coefficient and N is the normal force, and set it equal to the downhill component of weight, mg sin(θ), where m is the mass, g is the acceleration due to gravity, and θ is the angle of the incline. To prevent the cart from sliding, these forces must be equal. Hence, μs×N = mg sin(θ), and since N = mg cos(θ), we get μs = tan(θ). Therefore, the maximum angle, θ, can be found by taking the inverse tangent of the static friction coefficient, θ = tan-1(μs). This shows the relationship between the static friction coefficient and the maximum incline angle before sliding occurs.