First, let us analyze the problem and determine the governing equations for both cases.
Case (a) Negligible friction at slider B:
For this case, the tension in the cord (T) will be the force acting on the slider B due to its acceleration. First, we need to find the displacement of slider B (sB), which can be calculated from the given data.
sB = cord length - sA
sB = 1050 mm - 425 mm
sB = 625 mm
Now we need to determine the acceleration of the rod (aA) using the given data.
aA = rate of increase of the rod's speed
aA = 400 mm/s²
Since the cord length is constant, the acceleration of the slider B (aB) can be determined by multiplying the acceleration of the rod by the ratio of the lengths of the rod and the cord.
aB = aA * (sB / sA)
aB = 400 mm/s² * (625 mm / 425 mm)
aB = 588.24 mm/s²
Now we can calculate the tension in the cord using the mass of the slider B and its acceleration.
T = mass of slider B * acceleration of slider B
T = 0.5 kg * 588.24 mm/s²
T = 294.12 N
Therefore, the tension in the cord is 294.12 N for case (a).
Case (b) μk = 0.40 at slider B:
For this case, there will be a friction force (F_f) acting on the slider B in addition to the tension in the cord. We need to find the net force acting on the slider B (F_net) and then determine the tension in the cord accordingly.
First, let us determine the friction force using the given coefficient of kinetic friction (μk).
F_f = μk * mass of slider B * g
F_f = 0.40 * 0.5 kg * 9.81 m/s²
F_f = 1.962 N
Now we can determine the net force acting on the slider B.
F_net = mass of slider B * acceleration of slider B
F_net = 0.5 kg * 588.24 mm/s²
F_net = 294.12 N
Finally, we can calculate the tension in the cord for case (b) by subtracting the friction force from the net force acting on the slider B.
T = F_net - F_f
T = 294.12 N - 1.962 N
T = 292.16 N
Therefore, the tension in the cord is 292.16 N for case (b).
The rod of the fixed hydraulic cylinder is moving to the left with a speed of 100 mm/s and this speed is momentarily increasing at a rate of 400 mm/s each second at the instant when sA = 425 mm. Determine the tension in the cord at that instant. The mass of slider B is 0.5 kg, the length of the cord is 1050 mm, and the effects of the radius and friction of the small pulley at A are negligible. Find results for cases (a) negligible friction at slider B and (b) μk = 0.40 at slider B. The action is in a vertical plane.
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1 answer