Question

As a summer intern at UPS, you need to test the safety of a system of conveyor belts for moving packages. These are rubber sheets on which the boxes are moved from one place to another. Some of the belts are at an angle of 25° from the horizontal to move boxes up or down between the warehouse and the loading docks. Your boss thinks the systemm works well. You are concerned about boxes slipping if the belts are started too quickly. Your Experiment: You place a standard UPS brown box with a total mass of 10 kg on a stationary, horizontal conveyor belt. You attach a scale to the box and pull. * A force of 49 N is needed to start the box sliding if it is initially at rest. * It takes a force of 39 N to keep the box moving at a constant speed. What is your recommendation to your boss for the maximum acceleration of the belts? Note: The conveyor belts move horizontally, up, and down. You should consider each of these possibilities in your solution. The largest packages in the system are 55 kg a) GR & PS: Sketch the problem, both the Test and the operating conveyor belts that move the boxes. Define the important physical quantities in the problem. What information can you determine using your test? What models (and approximations) will you use? b) GR & AR: Draw any diagrams that will help you solve this problem. Write down force equations that match these diagrams. c) QC: Write down your known and unknown quantities in this problem. Determine if you have enough information to solve the problem. If you don't have enough equations, determine what other information you can use d) PS & AR: Set up and solve your equations for the information you need to determine your recommendation for the acceleration do this algebraically (analytically) . DO NOT USE NUMBERS e) QC: Check that your result for part (d) makes physical sense. Plug in numbers to get an answer for your boss

Answer 1

Given

Test mass m = 10 kg

Actual mass M = 55 kg

Static friction is 49 N

Kinetic friction is 39 N

Angle of of inclination 25^o

To find

Maximum acceleration

Solution

A)

WE can find the coefficients of static and kinetic friction from the test

F_s = μ_smg

49 = μ_s x 10 x 9.8

μ_s = 0.5

F_k = μ_kmg

39 = μ_k x 10 x 9.8

μ_k = 0.398

B)

When moving upward

Normal force = N = mgcosθ

F = mgsinθ + μ_smgcosθ

When moving downward

Normal force = N = mgcosθ

F + mgsinθ = μ_smgcosθ

C)

Known

μ_s = 0.5

Angle of inclination θ = 25^o

Acceleration due to gravity g = 9.8 m/s²

Unknown

Net force F = ma

We can solve for the value of a using the above equations

D)

When moving upward

Normal force = N = mgcos25

F = mgsin25 + μ_smgcos25

ma = mgsin25 + μ_smgcos25

a = gsin25 + μ_sgcos25

When moving downward

Normal force = N = mgcos25

F + mgsin25 = μ_smgcos25

ma + mgsin25 = μ_smgcos25

ma + mgsin25 = μ_smgcos25

ma = μ_smgcos25 - mgsin25

a = μ_sgcos25 - gsin25

E)

When moving upward

a = gsin25 + μ_sgcos25

a = 8.58 m/s²

When moving downward

a = μ_sgcos25 - gsin25

a = 0.299 m/s²

This makes sens. If the box is moving up we can give an acceleration upto 8.58 m/s²if it is moving down it should not be more than 0.299 m/s².