Newton's Law: Angled applied force and friction

[Starry]

Homework Statement

A boy is pushing a lawnmower (25kg) with 200N of force on a handle that is angled 35 below the horizontal. The mower is accelerating at 0.9m/s^2. If the boy would like to maintain a constant velocity, what force does he need to apply to the mower handle.
Assume g = 9.8 [d]

Homework Equations

F_net = ma
F_net = F_app + F_fr
F_fr = F_N * μ

The Attempt at a Solution

To find the components of the 200N force [35 degrees above horizon], I simply draw out the triangle and use sin and cos to find it. I found:
F_x = 200cos35 = 163.83N [forward]
F_y = 200sin35 = 114.71N [down]

Since it's accelerating, there is a net force. Using the two following equations,
F_net = ma
F_net = F_app + F_fr
(25)(0.9) = F_fr + F_x
(25)(0.9) = F_n * μ + 163.83
(25)(0.9) = (F_g + F_y) * μ + 163.83
and I end up with F_fr = 141.33N [backwards], and the coefficient of friction ends up to be 0.393.


To find the force required to keep the lawnmower in constant velocity: (Letting x2 and y2 be the new components of the second applied force)
F_fr2 = F_x2
F_N2 + F_y2 * μ = (245 + F_y2) * 0.393 = F_x2

I tried my best to show and explain what I have so far, but I don't know how to continue. I only have 1 equation and 2 unknowns. (The answer to the question is apparently 168N)

Thanks in advance.


EDIT:
I noticed a second relation that can be used: F_x2 * tan35 = F_y2
But when I solve the 2 unknowns using the 2 equations, I end up with;
F_x2 = 132.840N [forward]
F_y2 = 92.992N [down]
And if I use the Pythagorean theorem, I end up with F_app2 = 162N [35degrees below horizon], which is a little different than the answer of 168N.

 

[anathema]

Thank you for your post and for showing your work. Your approach is on the right track, but there are a few things that need to be corrected in order to get the correct answer of 168N.

First, when finding the components of the 200N force, you should use the sine and cosine functions with respect to the horizontal, not the vertical. This is because the angle given is measured from the horizontal, not the vertical. So the equations should be:

Fx = 200cos35 = 163.83N [forward]
Fy = 200sin35 = 114.71N [up]

Next, when setting up the equations for the net force, you should use the friction force as the unknown, not the net force. This is because the friction force is the force that is opposing the applied force and causing the acceleration. So the equations should be:

Ffr = ma - Fx
Ffr = μFN

Substituting in the values, we get:

Ffr = (25)(0.9) - 163.83
Ffr = 141.33N [backwards]

μFN = (25)(9.8)(0.393)
μFN = 96.135N

Now, to find the force required to maintain constant velocity, we can use the same equations, but with the acceleration set to 0 since the mower is not accelerating. So we get:

Ffr2 = Fx2
Ffr2 = μFN2

Substituting in the values, we get:

Ffr2 = 163.83 - Fx2
μFN2 = 96.135N

Solving for Fx2, we get:

Fx2 = 163.83 - 96.135 = 67.695N [forward]

Finally, to find the total force required to maintain constant velocity, we can use the Pythagorean theorem to find the magnitude of the force:

Fapp2 = √(Fx2^2 + Fy2^2)
Fapp2 = √(67.695^2 + 114.71^2)
Fapp2 = 168N

I hope this helps clarify your approach and leads you to the correct answer. Keep up the good work!