A 45ft long conveyor is used to move boxes up to a second level. The belt speed is 150ft/min and the conveyor is inclined at 15 degrees. Boxes are dropped vertically onto the conveyor and have no horizontal speed. The static and dynamic coefficient of friction between the belt and the box is u(s)=0.4 and u(k)=0.25 respectively. Determine the time it takes for a package to travel 45 ft on the conveyor.
To determine the time it takes for a package to travel 45 ft on the conveyor, we can divide the distance by the belt speed.
Given:
Conveyor length, L = 45 ft
Belt speed, V = 150 ft/min
Time, t = L / V
Substituting the given values:
t = 45 ft / 150 ft/min
Let's calculate the time:
To determine the time it takes for a package to travel 45 ft on the conveyor, we need to consider the system's dynamics involving the friction between the belt and the box.
First, let's break down the forces acting on the box on the inclined conveyor:
1. The gravitational force (mg), where m is the mass of the box and g is the acceleration due to gravity (approximately 32.2 ft/s^2).
2. The normal force (N), which acts perpendicular to the inclined surface of the conveyor.
3. The static friction force (fs), which opposes the motion of the box until it reaches its maximum value.
4. The kinetic friction force (fk), which comes into play once the box starts sliding.
To calculate the normal force, we need to decompose the weight force (mg) into its components parallel and perpendicular to the inclined surface. The perpendicular component is N = mg*cosθ, where θ is the angle of inclination (15 degrees).
The static friction force can be calculated as fs = u(s)*N, where u(s) is the static coefficient of friction.
The maximum value of static friction is given by fs(max) = u(s)*N. If the force required to move the box exceeds this value, the box will start sliding with a kinetic friction force fk = u(k)*N, where u(k) is the kinetic coefficient of friction.
In this case, since the box is dropped vertically and has no initial horizontal speed, the static friction force will initially oppose the motion of the box until it reaches the maximum value and the box starts sliding with kinetic friction.
Now, let's calculate the time it takes for the package to travel 45 ft on the conveyor:
1. Determine the acceleration of the box:
The net force acting on the box in the horizontal direction is given by F(net) = ma, where a is the acceleration.
The net force can be calculated by subtracting the friction force (fs or fk) from the horizontal component of the weight force, F(net) = m*g*sinθ - fs or fk.
In this case, since the box is initially at rest, the acceleration can be found by solving the equation a = F(net)/m.
2. Calculate the time it takes for the package to travel 45 ft:
We can use the equation of motion s = ut + (1/2)at^2, where s is the distance traveled, u is the initial velocity, t is the time, and a is the acceleration.
The initial velocity u is 0 because the box is initially at rest.
Rearranging the equation, we get t = √(2s/a).
By following these steps and plugging in the given values, we can determine the time it takes for a package to travel 45 ft on the conveyor.