You have been hired to help improve the material movement system at a manufacturing plant. Boxes containing 16 kg of tomato sauce in glass jars must slide from rest down a frictionless roller ramp to the loading dock, but they must not accelerate at a rate that exceeds 2.4 m/s^2 because of safety concerns.
Part A: What is the maximum angle of inclination of the ramp?
Express your answer to two significant figures and include appropriate units.
Part B: If the vertical distance the ramp must span is 1.4 m, with what speed will the boxes exit the bottom of the ramp?
Express your answer to two significant figures and include appropriate units.
Part C: What is the normal force on a box as it moves down the ramp?
Express your answer to two significant figures and include appropriate units.
If g= 9.81/s^2
then m g sin A = 9.81 m sin A = m a = m * 2.2
or in other words sin A = 2.2 / 9.81
kinetic energy at bottom = potential at top (there is no friction)
(1/2) v^2 = g h
v = sqrt (2 * 9.81 * 1.4)
normal force = m g cos A
typo, use 2.4 not 2.2
sin A = 2.4/9.81
To improve the material movement system at the manufacturing plant and ensure that the boxes of tomato sauce slide down the roller ramp safely, we need to determine the maximum angle at which the ramp can be inclined.
To find this angle, we can use the concept of forces and Newton's laws of motion. Let's break down the steps to calculate the maximum angle.
1. Calculate the maximum acceleration:
Since the boxes must not accelerate at a rate that exceeds 2.4 m/s^2, we can use this value as the maximum acceleration (a_max) of the system.
2. Determine the forces acting on the box:
The only force acting on the box while sliding down a frictionless ramp is the component of the weight force parallel to the ramp. This force is given by the formula:
F_parallel = m * g * sin(θ)
Where:
m = mass of the box (16 kg)
g = acceleration due to gravity (9.8 m/s^2)
θ = angle of inclination of the ramp
3. Equate the forces:
To prevent acceleration from exceeding 2.4 m/s^2, we need to set the force parallel to the ramp equal to the maximum force that the box can handle without exceeding this acceleration. We can calculate this by multiplying the mass of the box by the maximum acceleration:
F_parallel = m * a_max
m * g * sin(θ) = m * a_max
4. Solve for the maximum angle:
Rearrange the equation to solve for θ:
sin(θ) = a_max / g
θ = arcsin(a_max / g)
5. Calculate the maximum angle:
Plug in the values to calculate the maximum angle:
θ = arcsin(2.4 m/s^2 / 9.8 m/s^2)
θ ≈ 0.245 rad
θ ≈ 14.04°
Therefore, the ramp should not be inclined at an angle greater than approximately 14.04° to ensure that the boxes of tomato sauce do not accelerate at a rate exceeding 2.4 m/s^2. This will help improve safety in the material movement system at the manufacturing plant.