Question33. (II) A window washer pulls herself upward using the bucket-pulley apparatus shown in Fig. 4-40. (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 78 kg .
FIGURE $4-40$ Problem 33.

Studdy Solution

STEP 1

1. The window washer and the bucket together have a mass of 78 kg.
2. The window washer is using a pulley system to pull herself upward.
3. We assume the pulley is ideal (frictionless and massless).
4. We are trying to find the force needed to move at constant speed and the acceleration when the force is increased by 15%.

STEP 2

1. Analyze the forces involved when moving at constant speed.
2. Calculate the force needed to move at constant speed.
3. Analyze the forces when the force is increased by 15%.
4. Calculate the acceleration with the increased force.

STEP 3

Analyze the forces involved when moving at constant speed.
When moving at constant speed, the net force is zero. The force exerted by the window washer downward is equal to the gravitational force acting on her and the bucket.

STEP 4

Calculate the force needed to move at constant speed.
The gravitational force $F_g$ is given by:
$F_g = m \cdot g$
where $m = 78 \, \text{kg}$ is the mass of the person and bucket, and $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.
$F_g = 78 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 764.4 \, \text{N}$
Since the pulley system doubles the force exerted by the washer, the force she must exert downward $F_d$ is:
$F_d = \frac{F_g}{2} = \frac{764.4 \, \text{N}}{2} = 382.2 \, \text{N}$

STEP 5

Analyze the forces when the force is increased by 15%.
If the force is increased by 15%, the new force $F_{\text{new}}$ is:
$F_{\text{new}} = F_d + 0.15 \times F_d$

STEP 6

Calculate the acceleration with the increased force.
First, calculate the new force:
$F_{\text{new}} = 382.2 \, \text{N} + 0.15 \times 382.2 \, \text{N} = 439.53 \, \text{N}$
The net force $F_{\text{net}}$ is the difference between the new force and the gravitational force:
$F_{\text{net}} = 2 \times F_{\text{new}} - F_g$
$F_{\text{net}} = 2 \times 439.53 \, \text{N} - 764.4 \, \text{N}$
$F_{\text{net}} = 878.06 \, \text{N} - 764.4 \, \text{N} = 113.66 \, \text{N}$
Use Newton's second law to find the acceleration $a$ :
$F_{\text{net}} = m \cdot a$
$a = \frac{F_{\text{net}}}{m} = \frac{113.66 \, \text{N}}{78 \, \text{kg}}$
$a \approx 1.46 \, \text{m/s}^2$
The acceleration when the force is increased by 15% is approximately:
$\boxed{1.46 \, \text{m/s}^2}$

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