A window washer pulls herself upward using the bucket-pulley apparatus shown in Fig. 4-42. The mass of the person plus the bucket is 69 kg.How hard must she pull downward to raise herself slowly at constant speed?If she increases this force by 11 percent, what will her acceleration be?
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To find out how hard the window washer must pull downward to raise herself at a constant speed, we need to consider the forces acting on her.
First, let's analyze the forces when she is pulling downward with just enough force to counteract her weight (raising herself at a constant speed).
1. Weight force (mg): The force exerted by gravity on the person plus the bucket is given by the formula F = mg, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2). So, the weight force on the person plus the bucket is F = (69 kg)(9.8 m/s^2) = 676.2 N.
2. Tension force (T1): The person is pulling downward with a force equal in magnitude but opposite in direction to the weight force. Therefore, the tension force in the rope has the same magnitude as the weight force, which is 676.2 N.
Now, let's calculate the acceleration when the person increases her pulling force by 11 percent.
3. New tension force (T2): When she increases the force by 11 percent, the new tension force can be calculated as T2 = T1 + (11% of T1). So, T2 = 676.2 N + (0.11)(676.2 N) = 752.382 N.
4. Net force (Fnet): The net force on the person can be calculated by subtracting the new tension force from her weight force. Fnet = mg - T2 = (69 kg)(9.8 m/s^2) - 752.382 N = -77.982 N.
Since the window washer wants to rise at a constant speed, the net force should be zero (no acceleration). Therefore, she must pull downward with a force of 752.382 N to keep herself at a constant speed.
Finally, if the window washer increases her force by 11 percent, it means the new net force will no longer be zero, resulting in acceleration.
5. Acceleration (a): We can use Newton's second law of motion (F = ma) to calculate the acceleration. Since the net force (Fnet) and the mass (m) are known, we can rearrange the equation to solve for acceleration: a = Fnet/m.
Plugging in the values, a = -77.982 N / 69 kg = -1.129 m/s^2 (negative sign indicates the acceleration is in the opposite direction of the initial motion).
Therefore, if the window washer increases her downward pulling force by 11 percent, her acceleration will be -1.129 m/s^2.