n the figure provided, a 3.20 kg textbook is connected by a string to a 0.940 kg cup. The textbook is pushed up a slope of 24.3° with an initial speed of 4.29 m/s. If the coefficients of friction are μk = 0.350 and μs = 0.780 with the surface, how far will the textbook slide up the slope before stopping? Assume the rope and pulley are massless.
Weight of the cup = 1.23 * 9.8 = 12.054 N
Weight of textbook = 2.21 * 9.8 = 21.658 N
Force parallel = 21.658 * sin 22.3 ≈ 8.22 N
Friction force = 0.340 * 21.658 * cos 22.3 = 7.36372 * cos 22.3 ≈ 6.81 N
These last three forces are in the opposite direction of the red arrow.
Total force = 12.054 + 21.658 * sin 22.3 + 7.36372 * cos 22.3
This is approximately 27.09 N.
a = -(12.054 + 21.658 * sin 22.3 + 7.36372 * cos 22.3) ÷ 5.65
This is approximately -4.73 m/s^2. Let’s use the following equation to determine the distance it slides.
vf^2 = vi^2 + 2 * a * d
0 = 2.92^2 + 2 * [-12.054 + 21.658 * sin 22.3 + 7.36372 * cos 22.3) ÷ 5.65] * d
d = 8.5264÷ 2 * [-12.054 + 21.658 * sin 22.3 + 7.36372 * cos 22.3) ÷ 5.65]
This is approximately 0.9 meter.
Hope this helps.