Max distance up incline plane with variable friction

In summary, the conversation discusses finding the maximum distance a block moves up an incline at 30 degrees from the horizontal with an initial velocity of 10 m/s and a coefficient of friction that is 0.2 times the displacement. The solution involves using energy equations and taking into account the work done by the variable friction force. The final answer is 5.31 meters.
  • #1
PhizKid
477
1

Homework Statement


A block with initial velocity of 10 m/s is sent up an incline at 30 degrees from the horizontal. The coefficient of friction is 0.2x where x is the displacement. Find the maximum distance the block moves up the incline.


Homework Equations


F = ma
(1/2)mv^2
mgh

The Attempt at a Solution


I decided to try and use energy for this one.

Initial energy is (1/2)m(10)^2 + friction and final is mgh because the block comes to rest eventually.

So: (1/2)m(10)^2 + friction = mgh

Friction = coefficient * Normal force, so Friction = 0.2x * mgcos(30). Therefore:

(1/2)m(10)^2 + (0.2x*mgcos(30)) = mgh

Masses cancel out:

50 + (0.2x*gcos(30)) = gh

We want to find x because x is along the incline, and sin(30) = h / x, so h = xsin(30) and substitute back in:

50 + (0.2x*gcos(30)) = g(xsin30)

Simplify:

50 + 1.697x = 4.9x
50 = 3.203x
x = 15.6 meters

The solution is 5.31 meters, though. What have I done incorrectly?
 
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  • #2
PhizKid said:
So: (1/2)m(10)^2 + friction = mgh

By "friction" do you mean the force of friction or the work done by the friction force? Note that you are setting up an energy expression.
 
  • #3
Oh...the work done by friction would be 0.2x * mgcos(30) * x * cos(180) I think. So is that what I'm supposed to use for the "friction" part in my equation?
 
  • #4
PhizKid said:
Oh...the work done by friction would be 0.2x * mgcos(30) * x * cos(180) I think. So is that what I'm supposed to use for the "friction" part in my equation?

Right, you need the work done by friction. But the force of friction is a variable force (it depends on x). So, how do you get the work done by a variable force?
 
  • #5
Ohh I just learned this a few minutes ago, lol. Integrate the Friction Force, not multiply Friction Force * displacement.
 
  • #6
Yes. Good.
 

1. What is the formula for calculating the maximum distance up an incline plane with variable friction?

The formula for calculating the maximum distance is: d = μ * m * g * cosθ / (μ * cosθ + sinθ), where μ is the coefficient of friction, m is the mass of the object, g is the acceleration due to gravity, and θ is the angle of the incline.

2. How does the coefficient of friction affect the maximum distance up an incline plane?

The coefficient of friction plays a crucial role in determining the maximum distance up an incline plane. A higher coefficient of friction means there is more resistance to motion, resulting in a shorter distance. On the other hand, a lower coefficient of friction allows for less resistance and a longer distance.

3. Can the angle of the incline affect the maximum distance up an incline plane with variable friction?

Yes, the angle of the incline can affect the maximum distance. As the angle increases, the distance will decrease since there is more resistance to motion. This is because the component of gravity pulling the object down the incline increases as the angle increases.

4. What factors can affect the coefficient of friction on an incline plane?

The coefficient of friction on an incline plane can be affected by various factors such as the surface material of the incline and the object, the roughness of the surfaces, and the presence of any external forces acting on the object.

5. How is the maximum distance up an incline plane with variable friction useful in real-life applications?

The concept of calculating the maximum distance up an incline plane with variable friction is useful in many real-life applications. It can be used to determine the maximum angle at which a vehicle can safely travel on a road, the maximum height a ramp can have for wheelchairs, or the maximum angle at which a plane can take off on a runway.

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