An object with a mass of #3 kg# is on a ramp at an incline of #pi/12 #. If the object is being pushed up the ramp with a force of # 2 N#, what is the minimum coefficient of static friction needed for the object to remain put?

Question

An object with a mass of #3  kg# is on a ramp at an incline of #pi/12  #.  If the object is being pushed up the ramp with a force of # 2  N#, what is the minimum coefficient of static friction needed for the object to remain put?

Leo Abeyta · Accepted Answer

#mu_s >= 0.198#

We're asked to find the minimum coefficient of static friction #mu_s#.

For this situation, the net horizontal force #sumF_x# is given by

#sumF_x = mgsintheta - f_s - F_"applied" = 0#

where

#F_"applied"# is #2# #"N"# (directed up the incline)

#f_s# is the maximum static friction force (directed up the incline. Even though it is being pushed upward, the net force points downward in the absence of friction), which is given by

#f_s = mu_sn = mu_soverbrace(mgcostheta)^n#

Thus

#sumF_x = mgsintheta - mu_smgcostheta - 2# #"N"# #= 0#

Now we solve the expression for the coefficient of static friction #mu_s#:

#mu_smgcostheta = mgsintheta - 2# #"N"#

#ul(mu_s = (mgsintheta - 2color(white)(l)"N")/(mgcostheta)#

We are aware of:

#m = 3# #"kg"#

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

#theta = pi/12#

Putting these in gives

#mu_s = ((3color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)sin(pi/12) - 2color(white)(l)"N")/((3color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)cos(pi/12)) = color(blue)(ulbar|stackrel(" ")(" "0.198" ")|)#

The minimum coefficient of static friction is thus #color(blue)(0.198#.

William Audy · Answer

undefined The minimum coefficient of static friction needed for the object to remain put on the ramp can be calculated using the formula:

$$ \text{Coefficient of Static Friction} = \tan(\theta) $$

Where:
- $ \theta $ is the angle of the incline.

Given that the mass of the object is 3 kg and the force pushing it up the ramp is 2 N, we can calculate the force component parallel to the incline using the formula:

$$ \text{Force parallel to incline} = \text{Force pushing up} \times \sin(\theta) $$

Substituting the given values:

$$ \text{Force parallel to incline} = 2 \times \sin\left(\frac{\pi}{12}\right) $$

$$ \text{Force parallel to incline} ≈ 2 \times 0.2588 $$

$$ \text{Force parallel to incline} ≈ 0.5176 \, \text{N} $$

Now, we can calculate the force of gravity component parallel to the incline:

$$ \text{Force of gravity parallel to incline} = \text{mass} \times \text{gravity} \times \sin(\theta) $$

Substituting the given values:

$$ \text{Force of gravity parallel to incline} = 3 \times 9.8 \times \sin\left(\frac{\pi}{12}\right) $$

$$ \text{Force of gravity parallel to incline} ≈ 3 \times 9.8 \times 0.2588 $$

$$ \text{Force of gravity parallel to incline} ≈ 7.764 \, \text{N} $$

For the object to remain stationary, the force of static friction must counteract the force component parallel to the incline. Therefore, the minimum coefficient of static friction needed can be calculated using the formula:

$$ \text{Coefficient of Static Friction} = \frac{\text{Force of static friction}}{\text{Force of gravity parallel to incline}} $$

Substituting the given values:

$$ \text{Coefficient of Static Friction} = \frac{0.5176 \, \text{N}}{7.764 \, \text{N}} $$

$$ \text{Coefficient of Static Friction} ≈ 0.0667 $$

So, the minimum coefficient of static friction needed for the object to remain put on the ramp is approximately 0.0667.

Benjamin Asiedu · Answer

undefined The minimum coefficient of static friction needed for the object to remain put on the ramp can be calculated using the following steps:

1. Determine the gravitational force acting on the object parallel to the ramp. This is calculated as $ F_{\text{gravity}} = m \cdot g \cdot \sin(\theta) $, where $ m $ is the mass of the object, $ g $ is the acceleration due to gravity, and $ \theta $ is the angle of the incline.

2. Calculate the force component parallel to the ramp exerted by the applied force. This is given by $ F_{\text{applied}} = F_{\text{push}} \cdot \cos(\theta) $, where $ F_{\text{push}} $ is the applied force.

3. Determine the minimum static friction force needed to counteract the gravitational force and the applied force parallel to the ramp. This is expressed as $ F_{\text{friction}} = F_{\text{gravity}} + F_{\text{applied}} $.

4. Use the formula for static friction to find the minimum coefficient of static friction ($ \mu_{\text{static}} $). It can be calculated as $ \mu_{\text{static}} = \frac{F_{\text{friction}}}{N} $, where $ N $ is the normal force exerted by the ramp on the object, which is equal to $ m \cdot g \cdot \cos(\theta) $.

5. Substitute the calculated values into the formula to find $ \mu_{\text{static}} $.

6. Solve for $ \mu_{\text{static}} $.

Using these steps, you can find the minimum coefficient of static friction needed for the object to remain put on the ramp.