Sailors are using a manual capstan to pull an anchor chain. The capstan drum, around which… — Armed Services Vocational Aptitude Battery (ASVAB)

Sailors are using a manual capstan to pull an anchor chain. The capstan drum, around which the chain wraps, has a diameter of $1.6 \text{ feet}$ . The sailors push on the capstan bars at a horizontal distance of $4.0 \text{ feet}$ from the central axis of rotation. If the anchor chain exerts a steady pull of $1,500 \text{ pounds}$ , what is the total effort force the sailors must apply to the bars to keep the anchor moving at a constant speed? (Assume 100% efficiency and ignore friction)

A
$600 \text{ pounds}$
B
$7,500 \text{ pounds}$
$300 \text{ pounds}$ Cevap
D
$3,750 \text{ pounds}$

Cevap

300 \text{ pounds}

A capstan operates as a wheel and axle mechanism. The effort is applied at a distance of

4.0 \text{ feet}

(the wheel's radius), and the load is pulled by a drum with a radius of

0.8 \text{ feet}

(half of the

1.6\text{-foot}

diameter). The Ideal Mechanical Advantage is the ratio of these radii:

4.0 / 0.8 = 5

. To find the required effort force, divide the

1,500\text{-pound}

load by the mechanical advantage of

5

, which equals

300 \text{ pounds}

Adım Adım Çözüm

Determine the radii of both the wheel (effort arm) and the axle (drum).

Wheel radius (

R

) =

4.0 \text{ feet}

. Axle radius (

r

) =

1.6 \text{ feet} / 2 = 0.8 \text{ feet}

The Ideal Mechanical Advantage (IMA) of a wheel and axle depends on the ratio of their radii. The problem provides the diameter of the drum, which must be halved.

Calculate the Ideal Mechanical Advantage (IMA) of the capstan system.

IMA = \frac{R}{r} = \frac{4.0}{0.8} = 5

Because the effort is applied to the larger wheel (the bars), the system multiplies the input force by a factor of 5.

Calculate the required effort force to overcome the load.

\text{Effort Force} = \frac{\text{Load}}{IMA} = \frac{1500}{5} = 300 \text{ pounds}

The effort force is the total load divided by the mechanical advantage.

Anahtar Kavram

Ideal Mechanical Advantage of a Wheel and Axle

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