A truck pulls boxes up an incline plane. The truck can exert a maximum force of #5,000 N#. If the plane's incline is #pi/3 # and the coefficient of friction is #2/3 #, what is the maximum mass that can be pulled up at one time?

Question

A truck pulls boxes up an incline plane.  The truck can exert a maximum force of #5,000 N#.  If the plane's incline is #pi/3  # and the coefficient of friction is #2/3  #, what is the maximum mass that can be pulled up at one time?

Charles Aeschliman · Accepted Answer

#m_"max" = 425# #"kg"#

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 gravitational acceleration (acting down the incline), equal to #mgsintheta#

the friction force #f_k# (acting down the incline because it opposes motion)

the truck's upward pulling force

In the situation where the mass #m# is maximum, the boxes will be in equilibrium; i.e. the net force acting on them will be #0#.

We thus have our net force equation

#sumF_x = overbrace(F_"truck")^"upward" - overbrace(f_k - mgsintheta)^"downward" = 0#

The expression for the frictional force #f_k# is given by

#ul(f_k = mu_kn#

And since the normal force #color(green)(n = mgcostheta#, we can plug that in above:

#sumF_x = F_"truck" - overbrace(mu_kcolor(green)(mgcostheta))^(f_k = mu_kcolor(green)(n)) - mgsintheta = 0#

Or

#sumF_x = F_"truck" - mg(mu_kcostheta + sintheta) = 0#

Therefore,

#ul(F_"truck" = mg(mu_kcostheta + sintheta))color(white)(aaa)# (net upward force#=#net downward force)

And if we rearrange this equation to solve for the maximum mass #m#, we have

#color(red)(ulbar(|stackrel(" ")(" "m = (F_"truck")/(g(mu_kcostheta + sintheta))" ")|)#

The problem gives us

#F_"truck" = 5000# #"N"#

#mu_k = 2/3#

#theta = pi/3#

and #g = 9.81# #"m/s"^2#

Plugging in these values:

#m = (5000color(white)(l)"N")/((9.81color(white)(l)"m/s"^2)(2/3cos[(pi)/3] + sin[(pi)/3])) = color(blue)(ulbar(|stackrel(" ")(" "425color(white)(l)"kg"" ")|)#

Charlotte Aaron · Answer

undefined To find the maximum mass pulled up the incline, use the equation:

$$ m = \frac{F_{\text{max}} - f_{\text{friction}}}{g \cdot \sin(\theta) + \mu \cdot g \cdot \cos(\theta)} $$

Where:
$ F_{\text{max}} = 5,000 \, \text{N} $ (maximum force exerted by the truck)
$ f_{\text{friction}} = \mu \cdot m \cdot g $ (frictional force)
$ \theta = \frac{\pi}{3} $ (angle of the incline)
$ \mu = \frac{2}{3} $ (coefficient of friction)
$ g = 9.8 \, \text{m/s}^2 $ (acceleration due to gravity)

$$ m = \frac{5,000 - \frac{2}{3} \cdot m \cdot 9.8}{9.8 \cdot \sin\left(\frac{\pi}{3}\right) + \frac{2}{3} \cdot 9.8 \cdot \cos\left(\frac{\pi}{3}\right)} $$

Solving for $ m $ gives the maximum mass that can be pulled up.