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 2.6 m/s2 because of safety concerns. a. What is the maximum angle of inclination of the ramp?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?c. What is the normal force on a box as it moves down the ramp?

a)

b) 5.2 m/s

c) 151.2 N

Explanation:

a)

When the box is on the frictionless ramp, there is only one force acting in the direction along the ramp: the component of the forc of gravity parallel to the ramp, which is given by

where

m =16 kg is the mass of the box

is the acceleration due to gravity

is the angle of the ramp

According to Newton's second law of motion, the net force on the box is equal to the product of mass and acceleration, so:

where a is the acceleration.

From the equation above we get

And we are told that the acceleration must not exceed

Substituting this value and solving for , we find the maximum angle of the ramp:

b)

Here we are told that the vertical distance of the ramp is

Since there are no frictional forces acting on the box, the total mechanical energy of the box is conserved: this means that the initial gravitational potential energy of the box at the top must be equal to the kinetic energy of the box at the bottom of the ramp.

So we have:

where:

m = 16 kg is the mass of the box

h = 1.4 m height of the ramp

v = final speed of the box at the bottom of the ramp

Solving for v,

c)

There are two forces acting on the box in the direction perpendicular to the ramp:

- The normal force, N, upward

- The component of the weight perpendicular to the ramp, downward, of magnitude

Since the box is in equilibrium along the perpendicular direction, the net force is zero, so we can write:

and by substituting:

m = 16 kg

We can find the normal force:

at positions a, g kinetic energy is maximum.

at position d, potential energy is maximum.

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