11. A student pulls a 35 kg box up a ramp that is inclined at 12 degrees. If the box starts from rest at the bottom of the ramp and is pulled at 185 N at an angle of 25 degrees with respect to the incline, what is the acceleration of the box along the ramp? Assume the coefficient of kinetic friction is 0.27.
List of all the given values:
- mass, m = 35 kg
- angle of incline = 12°
- applied force, Fapp = 185 N, at angle = 25°
- coefficient of kinetic friction, μk = 0.27
From this, a force diagram can be drawn:
We can use trigonometry to find Fapp*x and Fapp*y:
- cos 25 = Fapp*x ÷ 185 N, Fapp*x = 185 N * cos 25 ≈ 167.667 N
- sin 25 = Fapp*y ÷ 185 N, Fapp*y = 185 N * sin 25 ≈ 78.184 N
The force of gravity Fg can be found from the equation:
- Fg = mg = 35 kg * 9.8 m/s2 = 343 N
Using this Fg value, we can also find the “x” and “y” components of Fg using trig.:
- sin 12 = Fg*x ÷ 343 N, Fg*x = 343 N * sin 12 ≈ 71.314 N
- cos 12 = Fg*y ÷ 343 N, Fg*y = 343 N * cos 12 ≈ 335.505 N
Now that we know all the “y” forces, we can calculate the FN using the Fnet*y equation:
- Fnet*y = FN + Fapp*y - Fg*y = may
- because there is no acceleration, Fnet*y = 35 kg * 0 = 0
- so, we can set FN + Fapp*y - Fg*y = 0
- fill in known values: FN + 78.184 N - 335.505 N = 0
- FN = 335.505 N - 78.184 N = 257.321 N
Using the FN and the coefficient of kinetic friction (given), we can calculate the Ff:
- Ff = μk * FN = 0.27 * 257.321 N = 69.477 N
Finally, now that we know all the “x” forces, we can calculate the Fnet*x. This can then be used to find the acceleration:
- Fnet*x = Fapp*x - Ff - Fg*x = max
- Fnet*x = 167.667 N - 69.477 N - 71.314 N = 26.876 N
- we can set this equal to max: max = 26.876
- plug in mass: 35 kg * ax = 26.876, ax ≈ 0.768 m/s2
The final answer to this problem is 0.768 m/s2.
