# A truck pulls boxes up an incline plane. The truck can exert a maximum force of #2,500 N#. If the plane's incline is #(3 pi )/8 # and the coefficient of friction is #1/3 #, what is the maximum mass that can be pulled up at one time?

We're asked to find the maximum mass that the truck is able to pull up the ramp, given the truck's maximum force, the ramp's angle of inclination, and the coefficient of friction.

The forces acting on the boxes are

the truck's upward pulling force

We thus have our net force equation

Or

Therefore,

The problem gives us

Plugging in these values:

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To find the maximum mass that can be pulled up the incline, we first need to determine the net force acting on the object. The force of gravity pulling the object down the incline is given by ( mg \sin(\theta) ), where ( m ) is the mass, ( g ) is the acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 )), and ( \theta ) is the angle of the incline. The force of friction opposing the motion is ( \mu N ), where ( \mu ) is the coefficient of friction and ( N ) is the normal force. The normal force is perpendicular to the incline and is equal to ( mg \cos(\theta) ).

Using the maximum force the truck can exert (2500 N) as the net force, we can set up the equation:

[ F_{\text{net}} = F_{\text{truck}} - F_{\text{friction}} - F_{\text{gravity}} = 0 ]

Substituting the expressions for each force:

[ 2500 , \text{N} - \frac{1}{3} \cdot mg \cdot \cos\left(\frac{3\pi}{8}\right) - mg \cdot \sin\left(\frac{3\pi}{8}\right) = 0 ]

Now, we can solve for the mass ( m ). After rearranging the equation and solving for ( m ), we get:

[ m = \frac{2500 , \text{N}}{g \cdot \left(\frac{1}{3} \cdot \cos\left(\frac{3\pi}{8}\right) + \sin\left(\frac{3\pi}{8}\right)\right)} ]

Plugging in the given values:

[ m = \frac{2500 , \text{N}}{9.8 , \text{m/s}^2 \cdot \left(\frac{1}{3} \cdot \cos\left(\frac{3\pi}{8}\right) + \sin\left(\frac{3\pi}{8}\right)\right)} ]

Calculating this expression will give us the maximum mass that can be pulled up the incline at one time.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- An object, previously at rest, slides #4 m# down a ramp, with an incline of #pi/4 #, and then slides horizontally on the floor for another #5 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
- The same force acts on two objects with different masses. Why does the object with less mass have a larger acceleration?
- If a #13 kg# object moving at #4 m/s# slows down to a halt after moving #80 m#, what is the friction coefficient of the surface that the object was moving over?

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