Chapter 4 Problem 32

Problem

A window washer pulls herself upward using the bucket–pulley apparatus

(a) How hard must she pull downward to raise herself slowly at constant speed?

(b) If she increases this force by 15%, what will her acceleration be? The mass of the person plus the bucket is 72 kg.

Solution

The following diagram shows only the forces on the bucket-washer combination, not forces exerted by the combination (the pull down on the rope by the person) or internal forces (normal force of bucket on person).

(a)

The net force must be zero if the bucket and washer have a constant speed.

$\sum F=F_{T}+F_{T}-mg=0$ $\rightarrow 2F_{T}=mg$ $\rightarrow F_{T}={\frac {1}{2}}mg={\frac {1}{2}}\left(72{\textrm {kg}}\right)\left(9.80{\textrm {m/s}}^{2}\right)=352.8{\textrm {N}}\approx 350{\textrm {N}}$

(b)

Now the force is increased by %15

$F_{T}=352.8{\textrm {N}}(1.15)=405.72{\textrm {N}}$

$\sum F=F_{T}+F_{T}-mg=ma$ $\rightarrow a={\frac {2F_{T}-mg}{m}}={\frac {2\left(405.72{\textrm {N}}\right)-\left(72{\textrm {kg}}\right)\left(9.80{\textrm {m/s}}^{2}\right)}{72{\textrm {kg}}}}=1.47{\textrm {m/s}}^{2}=1.5{\textrm {m/s}}^{2}$