A manual winch is used to lift a 240 \text{ lb} engine block exactly 2 \text{ feet} vertic… — Armed Services Vocational Aptitude Battery (ASVAB)

A manual winch is used to lift a $240 \text{ lb}$ engine block exactly $2 \text{ feet}$ vertically. The winch consists of a steering-wheel style control with a diameter of $3 \text{ feet}$ , which is rigidly attached to an axle drum with a radius of $3 \text{ inches}$ . Assuming an ideal system with no friction, what continuous effort force must be applied to the outer edge of the wheel, and what total distance must the effort force move along the wheel's rotation?

A
$20 \text{ lbs}$ of force; $24 \text{ feet}$ of distance
B
$480 \text{ lbs}$ of force; $1 \text{ foot}$ of distance
$40 \text{ lbs}$ of force; $12 \text{ feet}$ of distanceCevap
D
$1440 \text{ lbs}$ of force; $1/3 \text{ foot}$ of distance

Cevap

The correct answer is

40 \text{ lbs}

of force and

12 \text{ feet}

of distance.

The system's Ideal Mechanical Advantage is determined by the ratio of the effort radius (18 inches, half of the 3-foot diameter) to the load radius (3 inches), which yields an IMA of 6. This means the required effort force is divided by 6 (

240 \text{ lbs} / 6 = 40 \text{ lbs}

), while the required effort distance is multiplied by 6 (

2 \text{ feet} \times 6 = 12 \text{ feet}

) to conserve work.

Adım Adım Çözüm

Convert all measurements to the same unit (inches).

Wheel diameter =

3 \text{ feet} = 36 \text{ inches}

Units must be identical to calculate a valid, dimensionless mechanical advantage ratio.

Identify the effort radius and the load radius.

Wheel radius (

R

) =

18 \text{ inches}

. Axle drum radius (

r

) =

3 \text{ inches}

The lever arms in a wheel and axle system are the distances from the central axis (the radii), not the diameters.

Calculate the Ideal Mechanical Advantage (IMA).

\text{IMA} = \frac{R}{r} = \frac{18}{3} = 6

This establishes that the system multiplies the user's force by a factor of 6.

Calculate the required effort force.

\text{Effort Force} = \frac{\text{Load}}{\text{IMA}} = \frac{240 \text{ lbs}}{6} = 40 \text{ lbs}

Because the machine provides a mechanical advantage of 6, the required input force is one-sixth of the load.

Calculate the required effort distance.

\text{Effort Distance} = \text{Load Distance} \times \text{IMA} = 2 \text{ feet} \times 6 = 12 \text{ feet}

According to the principle of work, whatever factor force is reduced by, distance must be multiplied by that same factor.

Anahtar Kavram

Ideal Mechanical Advantage (IMA) and Work in a Wheel and Axle

Tahmini Süre:1m 30s

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