A box of sand (initially at rest) is pulled along a horizontal floor by a cable with a breaking strength of 1090.0 N. The coefficient of static friction between the box and the ground is 0.28. What should the angle between the cable and the floor be in order to move the greatest amount of sand?
To determine the angle between the cable and the floor that allows for the greatest amount of sand to be moved, we need to consider the forces acting on the box.
Let's break down the forces involved:
1. Weight (mg): This is the force due to gravity acting vertically downward. Its magnitude is given by the mass of the box (m) multiplied by the acceleration due to gravity (g).
2. Normal force (N): This is the force exerted by the floor on the box, perpendicular to the surface. It counteracts the weight of the box and is equal in magnitude but opposite in direction.
3. Static friction (Ff): This is the force that opposes the motion or impending motion between the box and the floor. Its magnitude is given by the coefficient of static friction (μs) multiplied by the normal force (N).
4. Tension in the cable (T): This is the force applied to pull the box. It has a maximum value of 1090.0 N, given by the breaking strength of the cable.
Now, let's analyze the forces acting in the horizontal direction. We want to find the angle at which the tension in the cable is maximum:
T - Ff = 0
Since Ff = μsN, we can rewrite the equation as:
T - μsN = 0
Now, let's analyze the forces acting in the vertical direction:
N - mg = 0
Solving this equation for N, we have:
N = mg
Substituting this into the previous equation, we get:
T - μsmg = 0
Now, we can solve this equation for the tension in the cable (T):
T = μsmg
Since T = 1090.0 N, we can rearrange the equation to solve for the angle θ between the cable and the floor:
μsmg = 1090.0 N
μs = 0.28 (given)
m = mass of the box
g = 9.8 m/s^2 (acceleration due to gravity)
Now, substitute these values into the equation and solve for θ:
0.28mg = 1090.0 N
θ = arctan(0.28mg / 1090.0)
The angle between the cable and the floor that allows for the greatest amount of sand to be moved is given by the arctan of the ratio (0.28mg / 1090.0). To find the specific value, you would need to know the mass of the box (m).