Simple Machines Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 5

Soru 81Soru →

A contractor uses a $15\text{-foot}$ ramp and a motorized winch to load a $600\text{-lb}$ generator. If the winch can pull with a maximum steady force of $120\text{ lbs}$ , what is the maximum vertical height, in feet, that the ramp can reach assuming a frictionless surface?

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Cevap: 3

Cevap

The maximum vertical height the ramp can reach is 3 feet.

An inclined plane multiplies the applied effort force by spreading the required work over a longer distance. First, determine the Mechanical Advantage needed by dividing the load (

600\text{ lbs}

) by the available effort force (

120\text{ lbs}

), which gives an IMA of

5

. This means the ramp must be

5

times longer than it is high. Given a ramp length of

15\text{ feet}

, divide this length by the IMA of

5

to find the maximum vertical height, which is

3\text{ feet}

.

Adım Adım Çözüm

1

Determine the required Ideal Mechanical Advantage (IMA).

\text{IMA} = \frac{600}{120} = 5

The IMA represents how many times the machine multiplies the effort force. It is found by dividing the load force by the effort force.

2

Use the inclined plane IMA formula to set up an equation for the height.

5 = \frac{15}{\text{Height}}

For an inclined plane, the

\text{IMA}

is also equal to the ratio of the

\text{Ramp Length}

to the

\text{Vertical Height}

.

3

Solve the equation for the vertical height.

\text{Height} = \frac{15}{5} = 3

Rearrange the equation to isolate the height, dividing the ramp length by the required IMA.

Anahtar Kavram

Ideal Mechanical Advantage (IMA) of an inclined plane and determining vertical height from force ratios.

Soru 82Soru →

A forestry worker uses a high-impact plastic felling wedge to direct the fall of a large tree. The wedge has a length (penetration depth) of $15\text{ cm}$ and a maximum heel thickness of $3\text{ cm}$ . If the worker strikes the wedge with a sledgehammer, applying an average effort force of $450\text{ N}$ , what is the theoretical maximum lifting force (in Newtons) exerted by the wedge? Assume $100\%$ efficiency.

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Cevap: 2250

Cevap

2250 Newtons

The correct theoretical maximum lifting force is found by first determining the mechanical advantage of the wedge (length divided by thickness), which is 5. Multiplying this mechanical advantage by the input effort force (450 N) yields an output force of 2250 N.

Adım Adım Çözüm

1

Identify the formula for the Ideal Mechanical Advantage (IMA) of a wedge.

\text{IMA} = \frac{\text{Length}}{\text{Thickness}}

The mechanical advantage of a wedge is determined by the ratio of its penetration length to its maximum separation thickness.

2

Calculate the IMA using the given dimensions.

\text{IMA} = \frac{15\text{ cm}}{3\text{ cm}} = 5

Substituting the length (

15\text{ cm}

) and thickness (

3\text{ cm}

) into the formula gives a mechanical advantage multiplier of 5.

3

Calculate the theoretical maximum output force.

\text{Output Force} = \text{IMA} \times \text{Input Force} = 5 \times 450\text{ N} = 2250\text{ N}

The output force is the product of the mechanical advantage and the applied effort force, assuming ideal conditions.

Anahtar Kavram

Ideal Mechanical Advantage (IMA) of Wedges

Soru 83Soru →

An electric motor directly turns the rear solid axle of an industrial transport cart. The axle has a diameter of $4 \text{ inches}$ , and the attached drive wheels have a diameter of $24 \text{ inches}$ . If the motor applies a rotational force of $1200 \text{ lbs}$ at the outer surface of the axle, what is the resulting forward force exerted by the wheels against the ground? (Assume zero friction losses).

200 \text{ lbs}

7200 \text{ lbs}

1200 \text{ lbs}

100 \text{ lbs}

Cevabı ve açıklamayı göster

Cevap: $200 \text{ lbs}$

Cevap

200 \text{ lbs}

In a vehicle drive system, the effort is applied to the smaller axle and the output is at the outer edge of the larger wheel. The mechanical advantage is the ratio of the input radius (

2 \text{ inches}

) to the output radius (

12 \text{ inches}

), which is

1/6

. Therefore, the

1200 \text{ lbs}

of input force is reduced to

1/6

of its value, resulting in

200 \text{ lbs}

of forward force.

Adım Adım Çözüm

1

Identify the input and output components of the machine.

Input is applied to the axle. Output is delivered at the edge of the wheel.

Determining where force is applied dictates whether the system acts as a force multiplier or a speed/distance multiplier.

2

Determine the radius of both the input and output components.

Axle radius =

2 \text{ inches}

(

4 \div 2

). Wheel radius =

12 \text{ inches}

(

24 \div 2

).

Mechanical advantage must be calculated using consistent dimensions, typically the radius of rotation.

3

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

\text{IMA} = \frac{\text{Input Radius}}{\text{Output Radius}} = \frac{2}{12} = \frac{1}{6}

.

When an axle drives a wheel, the output force is distributed over a larger distance, resulting in a mechanical advantage of less than 1.

4

Calculate the output force.

\text{Output Force} = \text{Input Force} \times \text{IMA} = 1200 \text{ lbs} \times \frac{1}{6} = 200 \text{ lbs}

.

The output force is the product of the applied effort and the mechanical advantage.

Anahtar Kavram

Mechanical Advantage of Axle-Driven Systems

Alternatif Yöntem

You can also use diameters directly without converting to radii: IMA = (Input Diameter) / (Output Diameter) = 4 / 24 = 1/6. Then multiply by the 1200 lbs force to get 200 lbs.

Tahmini Süre:1m 30s

Soru 84Soru →

A primary driving gear with 20 teeth operates at a speed of $120\text{ rpm}$ . Match each potential driven gear, identified by its tooth count, to the resulting output speed it would produce when directly meshed with this driving gear.

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Cevap

The 10-tooth gear matches to

240\text{ rpm}

, the 20-tooth gear matches to

120\text{ rpm}

, the 40-tooth gear matches to

60\text{ rpm}

, and the 60-tooth gear matches to

40\text{ rpm}

.

The correct matches are derived using the fundamental gear ratio principle: the speed of a driven gear is inversely proportional to its number of teeth compared to the driving gear.

Adım Adım Çözüm

1

Calculate the constant proportional value of the driving gear by multiplying its teeth by its speed.

20 \text{ teeth} \times 120\text{ rpm} = 2400

In a meshed gear system, the product of a gear's tooth count and its rotational speed remains constant for all meshing gears (

T_1 \times S_1 = T_2 \times S_2

).

2

Determine the output speed for each driven gear by dividing the constant value by its respective tooth count.

2400 \div 10 = 240

;

2400 \div 20 = 120

;

2400 \div 40 = 60

;

2400 \div 60 = 40

.

This isolates the unknown speed variable (

S_2

) for each specific gear configuration.

Anahtar Kavram

Gear Ratios and Speed Inverse Proportionality

Soru 85Soru →

Review the following gear train configurations. Match each specific gear arrangement to its resulting mechanical outcome or primary functional advantage.

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Cevap

Match the small driving gear to torque multiplication, the large driving gear to speed multiplication, the idler setup to identical direction/speed, and the worm gear to extreme reduction/self-locking.

Each arrangement fundamentally alters mechanical power in a distinct way: small-to-large increases torque, large-to-small increases speed, idlers are strictly for directional matching, and worm gears provide self-locking high-reduction.

Adım Adım Çözüm

1

Analyze the 15-tooth driving to 60-tooth driven gear arrangement.

It operates as a step-down gear system with a 1:4 ratio.

A smaller driving gear turning a larger driven gear inherently increases torque at the expense of rotational speed.

2

Analyze the 60-tooth driving to 15-tooth driven gear arrangement.

It operates as a step-up gear system with a 4:1 ratio.

A larger driving gear turning a smaller driven gear increases output rotational speed but reduces the available torque.

3

Analyze the two 30-tooth gears connected by an idler.

The output gear spins at the exact same speed as the input gear, and in the same rotational direction.

The intermediate idler gear reverses the rotational direction twice, ensuring input and output spin identically. The idler's size has no mathematical effect on the overall 1:1 ratio between the identical 30-tooth gears.

4

Analyze the worm and spur gear setup.

It drastically drops speed and cannot be easily back-driven.

A worm drive behaves like a continuous screw; a single full rotation of the worm may only advance the spur gear by one tooth, yielding massive reduction. Additionally, the angle typically prevents the spur gear from turning the worm.

Anahtar Kavram

Mechanical Advantage and Function of Gear Trains

Soru 86Soru →

Match each wheel and axle mechanism to its corresponding force-distance trade-off profile and Ideal Mechanical Advantage (IMA).

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Cevap

Screwdriver maps to IMA 6 (force multiplier). Bicycle maps to IMA 1/9 (distance multiplier). Steering wheel maps to IMA 14 (force multiplier). Ceiling fan maps to IMA 1/100 (speed multiplier).

Each mechanism is correctly paired by identifying where the effort is applied (wheel or axle) and calculating the ratio of the effort diameter to the load diameter. This ratio directly dictates whether the machine sacrifices distance for force (IMA > 1) or sacrifices force for distance/speed (IMA < 1).

Adım Adım Çözüm

1

Identify the effort and load components for each mechanism.

Determine which part acts as the 'wheel' (larger radius) and which acts as the 'axle' (smaller radius), and definitively locate where the input effort is applied.

The formula for Ideal Mechanical Advantage requires knowing whether the input is applied to the wheel or the axle.

2

Calculate the Ideal Mechanical Advantage (IMA) for each item.

Using the formula IMA = (Diameter of Effort) / (Diameter of Load). Screwdriver: 1.5/0.25 = 6. Bicycle: 3/27 = 1/9. Steering Wheel: 14/1 = 14. Ceiling Fan: 0.5/50 = 1/100.

This mathematical ratio represents the mechanical trade-off of the machine.

3

Classify each mechanism based on its IMA value.

IMA > 1 (Screwdriver, Steering Wheel) means the machine multiplies force. IMA < 1 (Bicycle, Ceiling Fan) means the machine multiplies distance or speed.

A machine cannot multiply both force and distance simultaneously due to the conservation of energy.

Anahtar Kavram

Ideal Mechanical Advantage (IMA) of a Wheel and Axle

Soru 87Soru →

A motor drives a small pinion gear with 10 teeth, which is meshed directly with a larger bull gear having 40 teeth. Compared to the small driving gear, which of the following correctly describes the output of the large driven gear?

It will rotate slower and produce more torque.

It will rotate faster and produce more torque.

It will rotate faster and produce less torque.

It will rotate slower and produce less torque.

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Cevap: It will rotate slower and produce more torque.

Cevap

The large driven gear will rotate slower and produce more torque compared to the small driving gear.

In a gear train, when a small gear drives a larger gear, the system is stepping down the speed. Because the driven gear has more teeth, it revolves less frequently than the driving gear. However, due to the conservation of energy (ignoring friction), this reduction in speed provides a mechanical advantage, meaning the output torque (turning force) is increased.

Adım Adım Çözüm

1

Analyze the size relationship between the driving gear and the driven gear.

The driving gear has 10 teeth, and the driven gear has 40 teeth. The driven gear is larger.

Gear size determines the mechanical advantage and the speed ratio.

2

Determine the effect of size on rotational speed.

Because the driven gear is 4 times larger (40/10 = 4), it will take 4 revolutions of the smaller gear to turn the larger gear once. Therefore, the driven gear rotates slower.

A smaller gear must turn multiple times to advance the many teeth of a larger gear through one full rotation.

3

Determine the effect on torque (turning force).

The decrease in speed is accompanied by a proportional increase in torque. The torque is multiplied by 4.

In simple machines like gear trains, trading distance (or speed) results in a gain in force (or torque), conserving mechanical power.

Anahtar Kavram

Mechanical Advantage in Gears

Soru 88Soru →

A $180 \text{ lb}$ load is suspended by a cable wrapped around a solid cylindrical axle. The cable hangs freely from the right side of the axle. A steering handwheel with a radius of $12 \text{ inches}$ is rigidly attached to this axle. The axle has a diameter of $8 \text{ inches}$ . To lift the load, a worker applies a tangential force to the outer edge of the handwheel. Viewed from the front, what is the required direction of rotation and the ideal effort force needed to lift the load at a constant speed?

Counterclockwise;

60 \text{ lbs}

Clockwise;

60 \text{ lbs}

Counterclockwise;

120 \text{ lbs}

Counterclockwise;

540 \text{ lbs}

Cevabı ve açıklamayı göster

Cevap: Counterclockwise; $60 \text{ lbs}$

Cevap

Counterclockwise;

60 \text{ lbs}

The correct answer accurately determines that raising a load from a cable hanging off the right side of a cylinder requires counterclockwise rotation. Furthermore, it accurately calculates the mechanical advantage by dividing the handwheel radius (

12 \text{ inches}

) by the axle radius (

4 \text{ inches}

), yielding an IMA of

3

. Dividing the

180 \text{ lb}

load by this IMA gives a required effort of

60 \text{ lbs}

.

Adım Adım Çözüm

1

Determine the required direction of rotation.

Counterclockwise rotation.

Since the cable hangs freely from the right side of the axle, pulling the right side upwards (winding the cable) requires turning the axle to the left, which is a counterclockwise direction.

2

Determine the working radii of the wheel and the axle.

Wheel radius (

R

) =

12 \text{ inches}

; Axle radius (

r

) =

4 \text{ inches}

.

The handwheel's radius is given as

12 \text{ inches}

. The axle's diameter is

8 \text{ inches}

, so its radius is half of that (

8 / 2 = 4 \text{ inches}

).

3

Calculate the Ideal Mechanical Advantage (IMA).

\text{IMA} = 3

For a wheel and axle system where effort is applied to the wheel,

\text{IMA} = R / r = 12 / 4 = 3

.

4

Calculate the required ideal effort force.

\text{Effort} = 60 \text{ lbs}

\text{Effort} = \text{Load} / \text{IMA} = 180 / 3 = 60 \text{ lbs}

.

Anahtar Kavram

Wheel and Axle Mechanical Advantage and Rotational Kinematics

Soru 89Soru →

Inside a wind-up toy, a main spring rotates a driving gear with 60 teeth at a speed of 10 RPM. This gear directly meshes with a smaller driven gear that powers the toy's legs. If the legs require the driven gear to spin at 40 RPM to function properly, how many teeth must the driven gear have?

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Cevap: 15

Cevap

15 teeth

The correct answer is derived using the principle that the product of the number of teeth and rotational speed must be equal for two meshed gears (

60 \times 10 = 15 \times 40

). This confirms the driven gear must have 15 teeth to rotate at 40 RPM.

Adım Adım Çözüm

1

Identify the relationship between gear teeth and rotational speed.

The governing formula is

T_1 \times S_1 = T_2 \times S_2

, where T represents the number of teeth and S represents the speed in RPM.

Gear speed is inversely proportional to the number of teeth.

2

Substitute the given numerical values into the equation.

60 \times 10 = T_2 \times 40

Applying the known driving gear parameters (60 teeth, 10 RPM) and the target speed for the driven gear (40 RPM).

3

Calculate the unknown number of teeth for the driven gear.

600 = T_2 \times 40

, which gives

T_2 = 15

Isolating the variable

T_2

yields the required tooth count for the mechanism to function correctly.

Anahtar Kavram

Gear Ratio and Speed Calculations

Soru 90Soru →

A carpenter uses a standard screwdriver to drive a screw into a piece of wood. The screwdriver handle, acting as the wheel, has a radius of $2 \text{ cm}$ . The metal shaft, acting as the axle, has a radius of $4 \text{ mm}$ . What is the ideal mechanical advantage (IMA) of the screwdriver?

5

0.5

2

0.2

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The ideal mechanical advantage is 5.

The correct answer is 5. To find the ideal mechanical advantage, both radii must first be in the same unit. Converting the

2 \text{ cm}

handle radius yields

20 \text{ mm}

. Dividing the wheel radius (

20 \text{ mm}

) by the axle radius (

4 \text{ mm}

) gives a mechanical advantage of 5, meaning the screwdriver multiplies the user's turning force by 5 times.

Adım Adım Çözüm

1

Convert the given measurements to the same unit (millimeters).

The handle (wheel) radius is

2 \text{ cm}

, which equals

20 \text{ mm}

.

Units must match before calculating any ratios to ensure accuracy.

2

Identify the values for the wheel and the axle.

Wheel radius =

20 \text{ mm}

; Axle radius =

4 \text{ mm}

.

In a screwdriver, the wider handle functions as the wheel, while the narrower shaft functions as the axle.

3

Calculate the Ideal Mechanical Advantage (IMA) by dividing the wheel radius by the axle radius.

IMA = 20 / 4 = 5

.

The formula for the IMA of a wheel and axle system driven by the wheel is the radius of the wheel divided by the radius of the axle.

Anahtar Kavram

Ideal Mechanical Advantage (IMA) of a Wheel and Axle

Soru 91Soru →

Match each gear arrangement with its resulting mechanical effect on speed, torque, or rotational direction.

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Cevap

A small driver to a large driven gear decreases speed and increases torque. A large driver to a small driven gear increases speed and decreases torque. Two directly meshed gears rotate in opposite directions. An idler gear between two gears allows the input and output to rotate in the same direction.

Each physical gear arrangement corresponds to a single, consistent mechanical law: size differences alter the speed/torque ratio inversely, while the number of meshing points dictates rotational direction.

Adım Adım Çözüm

1

Analyze the size relationship for the first two scenarios.

A smaller driver turning a larger gear trades speed for torque (gear reduction). A larger driver turning a smaller gear trades torque for speed (overdrive).

Understanding the fundamental trade-off between speed and torque is required to match size differences with output changes.

2

Analyze the rotational direction for direct mesh versus an idler train.

Direct mesh always reverses direction. Adding a single idler gear reverses it again, making the output match the input direction.

Counting the number of gears in a train determines the final direction of rotation.

Anahtar Kavram

Basic Gear Ratios and Rotational Direction

Tahmini Süre:1m 0s

Soru 92Soru →

A worker operates a well crank (a wheel and axle mechanism) to lift a heavy bucket of water. The crank handle travels in a circle with a radius of $12 \text{ inches}$ . The lifting rope is wound around an axle that has a diameter of $6 \text{ inches}$ . What is the Ideal Mechanical Advantage (IMA) of this system?

4

2

1

0.5

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The correct Ideal Mechanical Advantage (IMA) is

4

.

The correct answer is calculated by finding the ratio of the wheel's radius to the axle's radius. The wheel radius is given as

12 \text{ inches}

. The axle diameter is

6 \text{ inches}

, meaning its radius is

3 \text{ inches}

. Dividing

12

by

3

yields an IMA of

4

.

Adım Adım Çözüm

1

Identify the radius of the wheel (input distance).

The radius of the crank handle circle is

12 \text{ inches}

.

The IMA formula requires the radius of the wheel.

2

Calculate the radius of the axle (output distance).

The diameter is

6 \text{ inches}

, so the radius is

6 \div 2 = 3 \text{ inches}

.

The IMA formula requires the radius of the axle, not the diameter.

3

Calculate the Ideal Mechanical Advantage (IMA).

IMA = \frac{\text{Radius of Wheel}}{\text{Radius of Axle}} = \frac{12}{3} = 4

This ratio shows the mechanical advantage gained by applying force to the larger wheel to turn the smaller axle.

Anahtar Kavram

Ideal Mechanical Advantage of a Wheel and Axle

Tahmini Süre:45s

Soru 93Soru →

An industrial dough mixer is driven by a three-gear system. The motor is connected to Gear A (the driving gear), which has 15 teeth and rotates counter-clockwise at $300\text{ RPM}$ . Gear A meshes with an intermediate idler Gear B, which has 45 teeth. Gear B then meshes with Gear C (the final driven gear), which has 75 teeth and turns the mixer blade. What is the rotational speed and direction of Gear C?

60\text{ RPM}

, counter-clockwise

60\text{ RPM}

, clockwise

100\text{ RPM}

, counter-clockwise

1500\text{ RPM}

, counter-clockwise

Cevabı ve açıklamayı göster

Cevap: $60\text{ RPM}$ , counter-clockwise

Cevap

The driven gear will rotate at

60\text{ RPM}

in the counter-clockwise direction.

The correct answer accurately determines both speed and direction. In a simple gear train, the idler gear (Gear B) does not change the final speed ratio. The overall ratio is found by comparing the driving gear (15 teeth) to the final driven gear (75 teeth). The ratio is

15/75

, or

1/5

. Therefore, Gear C spins at

1/5

the speed of Gear A (

300 \times 1/5 = 60\text{ RPM}

). For direction, meshing gears alternate: Gear A (counter-clockwise) drives Gear B (clockwise), which in turn drives Gear C (counter-clockwise).

Adım Adım Çözüm

1

Determine the effect of the intermediate Gear B.

Gear B acts as an idler gear. It changes the direction of rotation between Gear A and Gear C, but does not affect the overall speed ratio of the system.

In a simple gear train, intermediate gears only bridge the gap and alter direction; the final speed is dictated exclusively by the first and last gears.

2

Calculate the final rotational speed of Gear C.

Speed ratio = (Teeth on Driving Gear) / (Teeth on Driven Gear) =

15 / 75 = 1/5

. The speed of Gear C is

300\text{ RPM} \times (1/5) = 60\text{ RPM}

.

Rotational speed is inversely proportional to the number of gear teeth. A larger driven gear will rotate slower than a smaller driving gear.

3

Determine the final direction of rotation for Gear C.

Gear A rotates counter-clockwise (CCW). Meshing Gear B rotates clockwise (CW). Meshing Gear C rotates counter-clockwise (CCW).

Adjacent meshing gears always rotate in opposite directions.

Anahtar Kavram

Idler gears change rotation direction but not the overall speed ratio, and larger driven gears result in a slower rotational speed.

Tahmini Süre:1m 0s

Soru 94Soru →

A mechanic is inspecting a steering mechanism. If the steering wheel has a radius of $14 \text{ inches}$ and the attached steering column (axle) has a radius of $2 \text{ inches}$ , what is the ideal mechanical advantage of this wheel and axle system?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The ideal mechanical advantage (IMA) is the ratio of the distance the effort moves to the distance the load moves. In a wheel and axle, this is proportional to their radii. Dividing the wheel's radius (

14 \text{ inches}

) by the axle's radius (

2 \text{ inches}

) gives an IMA of

7

.

Adım Adım Çözüm

1

Identify the relevant formula

\text{IMA} = \frac{R}{r}

The ideal mechanical advantage of a wheel and axle is the ratio of the effort radius (wheel) to the load radius (axle).

2

Substitute the known values

\text{IMA} = \frac{14}{2}

The wheel radius

R

is

14 \text{ inches}

and the axle radius

r

is

2 \text{ inches}

.

3

Perform the division

7

Dividing

14

by

2

yields an ideal mechanical advantage of

7

.

Anahtar Kavram

For a wheel and axle system where the input force is applied to the wheel, the ideal mechanical advantage is calculated by dividing the radius of the wheel by the radius of the axle.

Soru 95Soru →

A salvage crew is recovering a $3600 \text{ N}$ payload from a vertical shaft using a block and tackle system. The system consists of a fixed upper block with two sheaves and a movable lower block with two sheaves. The rope is anchored to the bottom of the fixed block, passes down to the first movable sheave, up to the first fixed sheave, down to the second movable sheave, up to the second fixed sheave, and finally is pulled downward by the crew.

The pulley system has a mechanical efficiency of $80\%$ .

If the crew needs to lift the payload by $3 \text{ meters}$ , what is the required actual effort force, and how many meters of rope must they pull?

Effort:

1125 \text{ N}

; Pull Distance:

12 \text{ m}

Effort:

900 \text{ N}

; Pull Distance:

15 \text{ m}

Effort:

1500 \text{ N}

; Pull Distance:

9 \text{ m}

Effort:

720 \text{ N}

; Pull Distance:

12 \text{ m}

Cevabı ve açıklamayı göster

Cevap: Effort: $1125 \text{ N}$ ; Pull Distance: $12 \text{ m}$

Cevap

The required actual effort force is

1125 \text{ N}

, and the crew must pull

12 \text{ m}

of rope.

The correct answer accurately determines that the system has an Ideal Mechanical Advantage (IMA) of 4. This is because there are four strands of rope directly pulling up on the movable block (the final downward pull only changes direction). This means the rope must be pulled 4 times the load distance (

4 \times 3\text{ m} = 12\text{ m}

). The ideal effort force is the load divided by 4 (

3600\text{ N} / 4 = 900\text{ N}

). Because the system is only 80% efficient, friction increases the required force. The actual effort force is calculated by dividing the ideal effort by the efficiency:

900\text{ N} / 0.80 = 1125\text{ N}

.

Adım Adım Çözüm

1

Determine the Ideal Mechanical Advantage (IMA) by counting the supporting rope strands.

There are 4 strands supporting the movable block. The 5th strand is pulled downward and does not support the load. Therefore, IMA = 4.

The IMA of a pulley system is equal to the number of rope segments that directly lift the movable block.

2

Calculate the required pull distance.

Pull Distance = Load Distance

\times

IMA =

3 \text{ m} \times 4 = 12 \text{ m}

.

The distance the effort must move is proportionally greater than the load distance based on the mechanical advantage.

3

Calculate the ideal effort force (assuming no friction).

Ideal Effort = Load / IMA =

3600 \text{ N} / 4 = 900 \text{ N}

.

In an ideal system, the force is reduced by the mechanical advantage factor.

4

Account for the system's mechanical efficiency to find the actual effort force.

Actual Effort = Ideal Effort / Efficiency =

900 \text{ N} / 0.80 = 1125 \text{ N}

.

Friction decreases efficiency, meaning more effort force is required than in an ideal system.

Anahtar Kavram

Calculating Mechanical Advantage, Effort Force, and Pull Distance in Inefficient Pulley Systems

Soru 96Soru →

A pipeline technician operates a circular handwheel to control the flow of water through a large valve. The handwheel has an overall diameter of $16 \text{ inches}$ and is attached to a central valve stem (the axle) that has a radius of $2 \text{ inches}$ . Assuming the effort force is applied to the outside edge of the handwheel, what is the ideal mechanical advantage (IMA) of this mechanism?

0.25

2

4

8

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Cevap: $4$

Cevap

The ideal mechanical advantage is

4

.

The ideal mechanical advantage (IMA) of a wheel and axle system is found by dividing the radius of the input force (the wheel) by the radius of the output force (the axle). Since the handwheel's diameter is

16 \text{ inches}

, its radius is

8 \text{ inches}

. Dividing the wheel radius (

8 \text{ inches}

) by the axle radius (

2 \text{ inches}

) gives an IMA of

4

.

Adım Adım Çözüm

1

Convert the handwheel diameter to a radius.

The handwheel radius is

16 \div 2 = 8 \text{ inches}

.

To calculate mechanical advantage accurately, both the wheel and the axle must use the same type of measurement (either both radii or both diameters).

2

Divide the wheel radius by the axle radius.

\text{IMA} = \frac{8 \text{ inches}}{2 \text{ inches}} = 4

.

The formula for the Ideal Mechanical Advantage of a wheel and axle (when force is applied to the wheel) is the radius of the wheel divided by the radius of the axle.

Anahtar Kavram

The ideal mechanical advantage of a wheel and axle is the ratio of the radius of the wheel to the radius of the axle.

Soru 97Soru →

Match each wheel and axle configuration to its resulting Ideal Mechanical Advantage (IMA).

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Cevabı ve açıklamayı göster

Cevap

Match each configuration by correctly dividing the effort radius by the load radius.

The correct matches are found by calculating the ratio of the input (effort) distance to the output (load) distance. For a wheel and axle, this is equivalent to

\frac{\text{Effort Radius}}{\text{Load Radius}}

. When the effort is on the wheel, the IMA is greater than 1. When the effort is on the axle, the IMA is less than 1.

Adım Adım Çözüm

1

Identify the effort radius and the load radius for each given configuration.

You determine whether the force is applied to the wheel (larger radius) or the axle (smaller radius).

The location of the applied force dictates which radius is the numerator in the IMA formula.

2

Apply the Ideal Mechanical Advantage formula:

\text{IMA} = \frac{\text{Effort Radius}}{\text{Load Radius}}

.

Calculations yield

5

,

4

,

0.5

, and

0.2

for the respective configurations.

This formula provides the theoretical multiplication of force for a wheel and axle mechanism.

Anahtar Kavram

Ideal Mechanical Advantage of a Wheel and Axle

Tahmini Süre:45s

Soru 98Soru →

A motorized telescope mount uses a compound gear system to adjust its viewing angle. The driving gear, Gear W, has 24 teeth and rotates counter-clockwise at $400\text{ RPM}$ . Gear W meshes directly with Gear X, which has 96 teeth. Gear X and Gear Y are rigidly mounted on the same intermediate shaft. Gear Y has 30 teeth and meshes with the final output gear, Gear Z, which has 120 teeth. What is the rotational speed and direction of Gear Z?

25\text{ RPM}

, counter-clockwise

25\text{ RPM}

, clockwise

400\text{ RPM}

, counter-clockwise

80\text{ RPM}

, clockwise

Cevabı ve açıklamayı göster

Cevap: $25\text{ RPM}$ , counter-clockwise

Cevap

The final output gear, Gear Z, rotates at

25\text{ RPM}

in the counter-clockwise direction.

In a compound gear system, the overall speed ratio is determined by analyzing each meshing pair. The first pair (W to X) reduces the speed from

400\text{ RPM}

to

100\text{ RPM}

(

400 \times 24/96

) and reverses the direction from counter-clockwise to clockwise. Since Gear Y shares a shaft with Gear X, it also rotates at

100\text{ RPM}

clockwise. The second meshing pair (Y to Z) further reduces the speed from

100\text{ RPM}

to

25\text{ RPM}

(

100 \times 30/120

) and reverses the direction again, making the final output counter-clockwise.

Adım Adım Çözüm

1

Calculate the rotational speed and direction of Gear X.

Speed of Gear X is

100\text{ RPM}

, rotating clockwise.

Gear X is driven by Gear W. The speed ratio is

24/96 = 1/4

.

400\text{ RPM} \times (1/4) = 100\text{ RPM}

. Meshing gears rotate in opposite directions, so Gear X rotates clockwise.

2

Determine the speed and direction of Gear Y.

Speed of Gear Y is

100\text{ RPM}

, rotating clockwise.

Gear Y is rigidly mounted on the same shaft as Gear X, meaning they share the exact same rotational speed and direction.

3

Calculate the rotational speed and direction of Gear Z.

Speed of Gear Z is

25\text{ RPM}

, rotating counter-clockwise.

Gear Z is driven by Gear Y. The speed ratio is

30/120 = 1/4

.

100\text{ RPM} \times (1/4) = 25\text{ RPM}

. Meshing gears rotate in opposite directions, so Gear Z rotates counter-clockwise.

Anahtar Kavram

Compound Gear Trains and Rotational Direction

Soru 99Soru →

A technician uses a T-handle socket wrench to tighten a nut. The wide horizontal T-handle functions as the wheel, and the narrow vertical shaft functions as the axle. If the technician turns the T-handle clockwise, what is the direction of the shaft's rotation, and how does the output force at the shaft compare to the effort force applied to the handle?

The shaft rotates clockwise, and the output force is greater than the effort force.

The shaft rotates counterclockwise, and the output force is greater than the effort force.

The shaft rotates clockwise, and the output force is less than the effort force.

The shaft rotates counterclockwise, and the mechanism acts as a first-class lever.

Cevabı ve açıklamayı göster

Cevap: The shaft rotates clockwise, and the output force is greater than the effort force.

Cevap

The shaft rotates clockwise, and the output force is greater than the effort force.

A wheel and axle mechanism consists of two parts that rotate together on the same axis. Therefore, if the wheel (T-handle) is turned clockwise, the axle (shaft) also turns clockwise. Because the input effort is applied to the larger radius of the handle, the mechanism multiplies the force, resulting in a greater output force at the smaller radius of the axle.

Adım Adım Çözüm

1

Determine the direction of rotation for the axle (shaft).

The shaft rotates clockwise, the same as the handle.

In a wheel and axle mechanism, both components are rigidly attached to the same axis and rotate as a single solid unit.

2

Evaluate the force relationship between the input and output.

The output force at the shaft is greater than the effort force applied to the handle.

Applying effort to the component with the larger diameter (the wheel/handle) to drive the component with the smaller diameter (the axle/shaft) creates a mechanical advantage greater than 1, which multiplies the force.

Anahtar Kavram

Wheel and Axle Rotational Direction and Mechanical Advantage

Soru 100Soru →

Wheel and axle mechanisms can be designed to multiply either force or speed. Match each real-world application of a wheel and axle to its correct mechanical description.

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Cevabı ve açıklamayı göster

Cevap

Doorknob matches the internal latch description; Automobile drive wheels match the engine and tire description; Manual hand winch matches the handle and cable drum description; Ceiling fan matches the motor and blades description.

Each mechanism uses the wheel and axle principle based on its specific need. The doorknob and hand winch are force multipliers because effort is applied to the larger wheel. The drive wheels and ceiling fan are speed multipliers because effort is applied to the smaller axle.

Adım Adım Çözüm

1

Identify the mechanisms that act as force multipliers (input on the wheel).

The doorknob and manual hand winch both receive input force on the larger radius to produce a larger output force on the smaller radius.

Determining the location of the input force helps classify the mechanical advantage.

2

Identify the mechanisms that act as speed multipliers (input on the axle).

The automobile drive wheels and ceiling fan both receive input force on the smaller central radius to produce greater speed at the larger outer radius.

Speed multipliers take advantage of the larger circumference of the wheel covering more distance in the same amount of time.

3

Pair each specific device with its corresponding functional description.

The doorknob operates an internal latch; the drive wheels rotate the outer tire; the winch winds a cable around a drum; the ceiling fan spins blades.

Matching the physical components completes the logical pairing for each wheel and axle application.

Anahtar Kavram

Force Multipliers vs. Speed Multipliers in Wheel and Axle Systems

Simple Machines

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