A 15kg box is resting on a hill forming an angle of a horizontal surface. The coefficient of static friction for the box on the surface is 0.45. Calculate the maximum angle of the incline just before the box starts to move.
Let A be the angle.
Friction force = Weight component down incline
M g * cos A * 0.45 = M g sinA
solve for A.
Mg cancels out. (Use arctangent function)
27
Both of the answers above are wrong answer is 24°
To calculate the maximum angle of the incline just before the box starts to move, we need to consider the forces acting on the box. The force of gravity acting vertically downwards can be split into two components: one parallel to the incline and the other perpendicular to the incline.
Now, let's break down the forces acting on the box:
1. Force of gravity (Fg): This force is acting vertically downwards and can be split into two components:
- Perpendicular component: Fg_perpendicular = mass (m) * gravitational acceleration (g) * cos(angle)
- Parallel component: Fg_parallel = mass (m) * gravitational acceleration (g) * sin(angle)
2. Normal force (Fn): This force is exerted by the surface and acts perpendicular to the incline.
3. Static friction force (Fs): This force opposes the motion of the box and acts parallel to the incline. The maximum static friction force before the box starts to move can be calculated as:
Fs_max = coefficient of static friction (µs) * Fn
For the box to be on the verge of moving, the parallel component of the force of gravity must equal the maximum static friction force:
Fg_parallel = Fs_max
Substituting the respective formulas, we have:
mass (m) * gravitational acceleration (g) * sin(angle) = coefficient of static friction (µs) * Fn
Since the normal force (Fn) is equal to the perpendicular component of the force of gravity, we can substitute by:
Fn = mass (m) * gravitational acceleration (g) * cos(angle)
Now, let's substitute the values we know and solve for the angle:
m * g * sin(angle) = µs * m * g * cos(angle)
In this equation, the mass (m), gravitational acceleration (g), and coefficient of static friction (µs) are known constants.
Simplifying the equation:
sin(angle) = µs * cos(angle)
To solve for the angle, we can take the arctan of both sides:
angle = arctan(µs)
Substituting the value of the coefficient of static friction (µs = 0.45):
angle = arctan(0.45)
Using a scientific calculator or trigonometric table, we can find the value of the arctan(0.45) to determine the maximum angle of the incline just before the box starts to move.