All practice questions

Question 2021Question →

A uniform steel bar is $10$ feet long and weighs $200$ pounds. It is placed on a pivot located exactly $2$ feet from its left end. To keep the bar perfectly horizontal, a downward force must be applied to the extreme left end. What is the magnitude of this downward force?

100

pounds

120

pounds

300

pounds

500

pounds

Show answer & explanation

Answer: $300$ pounds

Answer

The required downward force is

300

pounds.

To maintain a horizontal position, the torques around the pivot must balance perfectly. Because the bar is uniform, its entire

200

-pound weight acts at its center of gravity, which is

5

feet from the end. The pivot is located

2

feet from the left end, meaning the center of gravity is exactly

3

feet to the right of the pivot (

5 - 2 = 3

). This creates a clockwise torque of

600

lb-ft (

200

lbs

\times

3

ft). To balance this rotation, a downward force applied at the left end (which is

2

feet from the pivot) must create an equal

600

lb-ft counter-clockwise torque. Dividing the required

600

lb-ft by the

2

-foot distance gives a required downward force of

300

pounds.

Step-by-Step Solution

1

Determine the location of the center of gravity (CG).

The CG is exactly in the middle of the

10

-foot bar, which is

5

feet from either end.

A uniform object's weight is evenly distributed, so its total weight acts geometrically at its center.

2

Calculate the distances from the pivot point.

The left end is

2

feet from the pivot. The CG is

3

feet from the pivot (

5

feet -

2

feet).

Torque is calculated using the distance from the exact pivot (fulcrum) to the point where the force acts.

3

Calculate the clockwise torque created by the bar's weight.

200

lbs

\times

3

ft =

600

lb-ft.

The weight of the bar pulls down on the right side of the pivot, creating a clockwise rotational tendency.

4

Calculate the required counter-clockwise force at the left end.

Force

\times

2

ft =

600

lb-ft, which means Force =

300

lbs.

For the bar to remain horizontal (in equilibrium), the counter-clockwise torque must exactly balance the clockwise torque.

Key Concept

Principle of Moments (Torque Equilibrium)

Question 2022Question →

Match the following types of gear components to their primary mechanical function within a transmission system.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Rack and pinion maps to converting rotational to linear motion. Worm gear maps to providing drastic speed reduction and preventing back-driving. Bevel gear maps to changing the axis of rotation by 90 degrees. Idler gear maps to reversing rotational direction without changing the final output speed ratio.

Each gear type serves a distinct mechanical purpose: rack and pinions handle linear conversion, worm gears manage high reduction and locking, bevel gears handle angular power transmission, and idlers handle direction reversal without ratio changes.

Step-by-Step Solution

1

Identify the unique function of a rack and pinion system.

Matches with converting rotational motion into linear motion.

The circular pinion rolls along the flat rack, pushing it in a straight line, which is unique among these options.

2

Identify the primary characteristic of a worm gear drive.

Matches with providing drastic speed reduction and preventing back-driving.

The screw-like thread of the worm advances the gear by one tooth per revolution, giving high reduction, and friction usually prevents reverse operation.

3

Identify the application of a bevel gear set.

Matches with changing the axis of rotation by 90 degrees between intersecting shafts.

The conical shape of bevel gears allows them to mesh at an angle, most commonly 90 degrees, allowing perpendicular power transmission.

4

Identify the purpose of an idler gear.

Matches with reversing the rotational direction without changing the final output speed ratio.

An intermediate gear only changes the spin direction of the subsequent gear; its tooth count cancels out in the overall ratio calculation.

Key Concept

Functions of specialized gear types in power transmission

Question 2023Question →

Two outdoor swimming pools are filled with fresh water to an exact depth of $3\text{ meters}$ . The first is a small backyard wading pool, while the second is a massive Olympic-sized pool holding a much larger total volume of water. How does the hydrostatic pressure at the very bottom of the backyard pool compare to the hydrostatic pressure at the bottom of the Olympic-sized pool?

The pressure is exactly equal in both pools because they share the same fluid density and water depth.

The pressure is significantly greater in the Olympic-sized pool because it contains a larger total mass of water.

The pressure at the bottom of both pools is exactly

3\text{ Pascals}

because the depth is exactly

3\text{ meters}

.

The pressure depends entirely on the buoyant force, which requires a heavy object to be placed in the water.

Show answer & explanation

Answer: The pressure is exactly equal in both pools because they share the same fluid density and water depth.

Answer

The pressure is exactly equal in both pools because they share the same fluid density and water depth.

The correct answer accurately states that hydrostatic pressure relies solely on depth and fluid density. Because both pools contain fresh water and are filled to the exact same depth of

3\text{ meters}

, the pressure at the bottom of both pools is identical, despite their vast difference in total volume.

Step-by-Step Solution

1

Identify the factors that determine hydrostatic pressure in a static fluid.

Hydrostatic pressure is determined by the fluid's density, the acceleration due to gravity, and the depth of the fluid.

To evaluate whether factors like total volume or container shape influence the final pressure reading.

2

Compare the relevant factors between the two distinct swimming pools.

Both pools contain fresh water (same density) and have an exact depth of

3\text{ meters}

.

To determine if there are any differences in the variables that actually calculate pressure.

3

Deduce the relative pressure at the bottom of each pool based on the identified factors.

Since both fluid density and depth are identical, the hydrostatic pressure at the bottom must be identical.

According to the principles of fluid mechanics, hydrostatic pressure is independent of the container's width or the fluid's total mass.

Key Concept

Hydrostatic pressure is dependent only on fluid density and depth, not on the container's shape or the total volume of fluid.

Question 2024Question →

A military hovercraft with a mass of $6,000 \text{ kg}$ is transporting equipment across a frozen lake. The hovercraft's thrusters generate a forward force of $27,000 \text{ N}$ , while air resistance and surface friction create a combined opposing force of $9,000 \text{ N}$ . What is the forward acceleration of the hovercraft?

3.0 \text{ m/s}^2

4.5 \text{ m/s}^2

0.3 \text{ m/s}^2

0.33 \text{ m/s}^2

Show answer & explanation

Answer: $3.0 \text{ m/s}^2$

Answer

The correct forward acceleration of the hovercraft is

3.0 \text{ m/s}^2

.

The correct answer accurately determines the net force by subtracting the opposing friction (

9,000 \text{ N}

) from the applied thrust (

27,000 \text{ N}

) to get

18,000 \text{ N}

. Dividing this net force by the mass (

6,000 \text{ kg}

) yields an acceleration of

3.0 \text{ m/s}^2

.

Step-by-Step Solution

1

Calculate the net forward force acting on the hovercraft.

F_{net} = 27,000 \text{ N} - 9,000 \text{ N} = 18,000 \text{ N}

Acceleration is determined by the net force, which is the sum of all forces acting on the object. The opposing force must be subtracted from the forward thrust.

2

Set up Newton's Second Law equation.

F_{net} = m \times a

This formula relates net force, mass, and acceleration.

3

Rearrange the equation to solve for acceleration and substitute the known values.

a = \frac{18,000 \text{ N}}{6,000 \text{ kg}} = 3.0 \text{ m/s}^2

Dividing the net force by the mass gives the resulting acceleration.

Key Concept

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass (

a = \frac{F_{net}}{m}

).

Question 2025Question →

A uniform $10$ -foot wooden beam weighing $50$ pounds is placed on a flat surface. In four different scenarios, exactly $150$ pounds of additional weight is placed on the beam in various configurations. Arrange the scenarios in order of the beam's overall balance point (center of gravity), measured from the LEFT end of the beam, from the GREATEST distance to the LEAST distance.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from greatest to least distance from the left end is: the two

75

-pound weights (furthest right), the

50

/

100

-pound weights, the

100

/

50

-pound weights, and finally the single

150

-pound weight (closest to the left).

Because the total system weight is identical (

200

lbs) in every scenario, the relative position of the center of gravity is determined purely by the sum of the moments (weight

\times

distance) of the added weights from the left end. Calculating these moments yields

1050

,

900

,

600

, and

450

ft-lbs respectively. A larger moment means the center of gravity is pulled further to the right, which translates to a greater overall distance from the left end.

Step-by-Step Solution

1

Identify the components that determine the overall center of gravity (balance point).

The center of gravity's distance from an origin (the left end) is found by dividing the total turning moment (torque) by the total weight of the system.

This is the fundamental principle of moments for finding the balance point of a loaded beam.

2

Observe the constants across all four scenarios.

In every scenario, the total system weight is identical at

200

lbs (

50

lb beam +

150

lbs of added weight), and the beam's own moment from the left end remains constant.

Because the total weight denominator is identical across all choices, the relative position of the balance point is determined solely by the sum of the moments of the added weights.

3

Calculate the turning moment (Weight

\times

Distance) of the added weights relative to the left end for each scenario.

Two

75

-lb weights:

(75 \times 4) + (75 \times 10) = 1050

ft-lbs.

50

-lb and

100

-lb weights:

(50 \times 2) + (100 \times 8) = 900

ft-lbs.

100

-lb and

50

-lb weights:

(100 \times 2) + (50 \times 8) = 600

ft-lbs.
Single

150

-lb weight:

150 \times 3 = 450

ft-lbs.

This calculation provides a numerical value directly proportional to how far right the weight is biased.

4

Order the scenarios by their added moments from greatest to least.

The order is

1050 > 900 > 600 > 450

. Therefore, the balance point distances follow the exact same order.

A greater turning moment from the left end means the center of gravity is located further to the right, which translates to a greater distance from the left end.

Key Concept

The center of gravity of a system shifts proportionally toward the location with the greatest turning moment (Force

\times

Distance). When total mass is constant, ranking moments ranks the balance points.

Estimated Time:1m 30s

Question 2026Question →

A sledgehammer head with a weight of $19.6 \text{ N}$ on Earth strikes a metal spike with a downward velocity of $4 \text{ m/s}$ . The impact brings the hammer head to a complete stop in $20 \text{ ms}$ (milliseconds). What is the average impact force exerted on the spike? (Assume $g = 9.8 \text{ m/s}^2$ )

0.4 \text{ N}

3,920 \text{ N}

400 \text{ N}

0.16 \text{ N}

Show answer & explanation

Answer: $400 \text{ N}$

Answer

The correct answer is

400 \text{ N}

.

The force is found by first converting weight to mass (

19.6 / 9.8 = 2 \text{ kg}

) and time to seconds (

20 \text{ ms} = 0.02 \text{ s}

). The change in momentum is

2 \text{ kg} \times 4 \text{ m/s} = 8 \text{ kg} \cdot \text{m/s}

. Dividing this change by the impact duration (

0.02 \text{ s}

) gives an average impact force of

400 \text{ N}

.

Step-by-Step Solution

1

Calculate the mass of the sledgehammer head.

m = \frac{W}{g} = \frac{19.6 \text{ N}}{9.8 \text{ m/s}^2} = 2 \text{ kg}

Momentum equations require mass, not weight.

2

Convert the impact time to standard SI units (seconds).

t = 20 \text{ ms} = 0.02 \text{ s}

Standard units must be used to calculate force in Newtons.

3

Calculate the change in momentum (impulse).

\Delta p = m \times \Delta v = 2 \text{ kg} \times 4 \text{ m/s} = 8 \text{ kg} \cdot \text{m/s}

Impulse is equal to the change in momentum.

4

Calculate the average impact force.

F = \frac{\Delta p}{\Delta t} = \frac{8 \text{ kg} \cdot \text{m/s}}{0.02 \text{ s}} = 400 \text{ N}

Force is the rate of change of momentum.

Key Concept

Impulse and Momentum

Question 2027Question →

A heavy-duty diving bell is being tested in a deep saltwater facility. At a depth of $10\text{ meters}$ , the hydrostatic pressure exerted on its exterior is measured to be $100\text{ kilopascals (kPa)}$ . If the diving bell is lowered further to a depth of $30\text{ meters}$ , what will the new hydrostatic pressure be in $\text{kPa}$ , assuming the fluid density remains constant?

Show answer & explanation

Answer: 300

Answer

The new hydrostatic pressure will be 300 kPa.

Because hydrostatic pressure is directly proportional to depth in a fluid of constant density, tripling the depth from 10 meters to 30 meters results in exactly three times the pressure. Thus, 100 kPa multiplied by 3 equals 300 kPa.

Step-by-Step Solution

1

Identify the relationship between hydrostatic pressure and depth.

Pressure is directly proportional to depth.

This is a fundamental principle of hydrostatic pressure for a fluid with constant density.

2

Calculate the ratio of the new depth to the initial depth.

30\text{ m} / 10\text{ m} = 3

This ratio tells us how much the pressure will increase.

3

Calculate the new pressure.

100\text{ kPa} \times 3 = 300\text{ kPa}

Applying the direct proportion to find the final pressure value.

Key Concept

Hydrostatic pressure increases in direct proportion to the depth of the fluid, assuming fluid density remains constant.

Question 2028Question →

Arrange the following vehicle scenarios in order from the lowest acceleration to the highest acceleration.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from lowest to highest acceleration is: the forklift (

0 \text{ m/s}^2

), the motorized cart (

2 \text{ m/s}^2

), the delivery van (

4 \text{ m/s}^2

), and the sports car (

5 \text{ m/s}^2

).

By applying the acceleration formula

a = \frac{\Delta v}{t}

to each scenario, we find the respective accelerations are

0 \text{ m/s}^2

,

2 \text{ m/s}^2

,

4 \text{ m/s}^2

, and

5 \text{ m/s}^2

. Placing them in ascending numeric order yields the correct sequence.

Step-by-Step Solution

1

Calculate the acceleration of the forklift.

a = 0 \text{ m/s}^2

Because the forklift maintains a constant speed in a straight line, its velocity does not change.

2

Calculate the acceleration of the motorized cart.

a = 2 \text{ m/s}^2

Using the formula

a = \frac{v_f - v_i}{t}

, we get

\frac{6 - 0}{3} = 2

.

3

Calculate the acceleration of the delivery van.

a = 4 \text{ m/s}^2

Using the formula

a = \frac{v_f - v_i}{t}

, we get

\frac{22 - 10}{3} = 4

.

4

Calculate the acceleration of the sports car.

a = 5 \text{ m/s}^2

Using the formula

a = \frac{v_f - v_i}{t}

, we get

\frac{15 - 0}{3} = 5

.

Key Concept

Acceleration is the rate at which an object changes its velocity, calculated by dividing the change in velocity by the elapsed time.

Estimated Time:1m 0s

Question 2029Question →

Two solid machine components, Component X and Component Y, have the exact same volume but are cast from different metals. Component X weighs $15 \text{ N}$ in air, and Component Y weighs $45 \text{ N}$ in air.

Both components are separately lowered into identical displacement tanks filled to the brim with water. When Component X is completely submerged, exactly $5 \text{ N}$ of water spills out of the tank. When Component Y is completely submerged, what is the weight of the water that spills out?

5 \text{ N}

15 \text{ N}

40 \text{ N}

45 \text{ N}

Show answer & explanation

Answer: $5 \text{ N}$

Answer

The weight of the water that spills out is

5 \text{ N}

.

The correct answer is

5 \text{ N}

. When an object is fully submerged in a fluid, it displaces a volume of fluid equal to its own volume. According to Archimedes' Principle, the buoyant force acting on the object is equal to the weight of this displaced fluid. Because Component Y has the exact same volume as Component X, it must displace the exact same amount of water, meaning the weight of the overflowing water remains

5 \text{ N}

. The object's weight in air does not affect how much fluid it displaces once it is completely submerged.

Step-by-Step Solution

1

Identify the relationship between the two components.

Both Component X and Component Y share the exact same volume, despite having different weights in air.

The physical volume of an object dictates how much fluid it can displace when fully submerged.

2

Apply Archimedes' principle of displacement.

Because both components have identical volumes, completely submerging either one will displace the exact same volume of water.

A fully submerged object always displaces a volume of fluid equal to its own total volume, regardless of its density or mass.

3

Determine the weight of the displaced water for Component Y.

Since Component X displaced

5 \text{ N}

of water, Component Y must also displace

5 \text{ N}

of water.

Equal volumes of the same fluid (water) will naturally have the same weight.

Key Concept

Buoyancy and Archimedes' Principle

Estimated Time:1m 0s

Question 2030Question →

In heavy steel construction, solid steel rivets are often heated to a red-hot state before being inserted into structural holes and hammered into place. Aside from making the metal softer and easier to shape, what is the primary mechanical advantage gained from the thermal properties of the rivet?

As the rivet cools to ambient temperature, it expands, wedging itself immovably within the hole.

The high temperature increases the overall mass of the rivet, providing a stronger downward force on the joint.

As the rivet cools to ambient temperature, it contracts, pulling the steel plates tightly together.

The heat transfers into the surrounding steel plates, causing them to contract and securely grip the rivet shaft.

Show answer & explanation

Answer: As the rivet cools to ambient temperature, it contracts, pulling the steel plates tightly together.

Answer

As the rivet cools to ambient temperature, it contracts, pulling the steel plates tightly together.

When the red-hot steel rivet cools down to ambient temperature, it undergoes thermal contraction. Since the ends of the rivet are hammered flat against the outer surfaces of the plates during installation, the decreasing length of the cooling rivet shaft creates a powerful tension that pulls the plates tightly together, creating a highly secure joint.

Step-by-Step Solution

1

Analyze the thermal state of the rivet during installation.

The rivet is installed while red-hot, meaning it is at its maximum thermal expansion.

Understanding the initial state is required to predict the dimensional changes as the system returns to equilibrium.

2

Apply the principle of thermal expansion and contraction to the cooling phase.

As the hot rivet cools down to the temperature of the surrounding environment, the metal will contract (shrink).

Most solid materials, including structural steel, expand when heated and contract when cooled.

3

Evaluate the mechanical effect of this contraction on the structural joint.

Because the rivet heads are formed on the outside of the plates while hot, the subsequent contraction of the rivet shaft pulls the plates firmly against one another.

The length of the rivet attempts to decrease, creating a strong clamping force (tension) between the two heads that locks the plates in place.

Key Concept

Thermal Contraction and Mechanical Clamping Force

Question 2031Question →

A conveyor system uses a compound gear train to reduce motor speed. The driving motor turns Gear A (20 teeth) clockwise at $240\text{ rpm}$ . Gear A meshes with Gear B (60 teeth). Gear B shares a solid axle with Gear C (10 teeth), meaning they rotate together. Gear C meshes with Gear D (40 teeth), which drives the conveyor belt. What is the rotational speed and direction of Gear D?

20\text{ rpm}

, clockwise

20\text{ rpm}

, counter-clockwise

120\text{ rpm}

, counter-clockwise

2880\text{ rpm}

, clockwise

Show answer & explanation

Answer: $20\text{ rpm}$ , clockwise

Answer

Gear D rotates at

20\text{ rpm}

, clockwise.

In a compound gear system, the total speed reduction is the product of the individual gear pair ratios. The first pair reduces the speed from

240\text{ rpm}

to

80\text{ rpm}

and reverses direction to counter-clockwise. The second pair reduces the

80\text{ rpm}

speed to

20\text{ rpm}

and reverses direction back to clockwise.

Step-by-Step Solution

1

Calculate the speed and direction of Gear B.

Gear B rotates at

80\text{ rpm}

counter-clockwise.

Gear B has three times as many teeth as Gear A (60 vs 20), so it spins at one-third the speed (

240 / 3 = 80

). Meshing gears turn in opposite directions.

2

Determine the speed and direction of Gear C.

Gear C rotates at

80\text{ rpm}

counter-clockwise.

Gears mounted on the same solid axle rotate at the exact same speed and in the same direction.

3

Calculate the speed and direction of Gear D.

Gear D rotates at

20\text{ rpm}

clockwise.

Gear D has four times as many teeth as Gear C (40 vs 10), so it spins at one-quarter the speed of Gear C (

80 / 4 = 20

). Meshing gears turn in opposite directions.

Key Concept

Compound Gear Trains and Speed Reduction

Question 2032Question →

Match each flight deck towing and movement scenario to its resulting acceleration using Newton's Second Law ( $F = ma$ ). Assume no friction unless stated.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair the aircraft with

0.8 \text{ m/s}^2

, the munitions cart with

4.0 \text{ m/s}^2

, the helicopter with

0.25 \text{ m/s}^2

, and the fuel pod with

1.2 \text{ m/s}^2

.

Each acceleration is determined by dividing the net force acting on the object by its mass. In scenarios where multiple forces act on the object, the net force must be found first by subtracting any resistance from the applied force.

Step-by-Step Solution

1

Calculate the acceleration for the aircraft.

Dividing the net force of

11,200 \text{ N}

by the mass of

14,000 \text{ kg}

yields

0.8 \text{ m/s}^2

.

Applying Newton's Second Law directly.

2

Calculate the acceleration for the munitions cart.

Dividing the net force of

12,800 \text{ N}

by the mass of

3,200 \text{ kg}

yields

4.0 \text{ m/s}^2

.

Applying Newton's Second Law directly.

3

Determine the net force and calculate the acceleration for the helicopter.

Subtract the

2,000 \text{ N}

resistance from the

6,000 \text{ N}

applied force to find a net force of

4,000 \text{ N}

. Dividing this by the

16,000 \text{ kg}

mass gives

0.25 \text{ m/s}^2

.

Newton's Second Law requires the net force, which is the sum of all opposing forces.

4

Calculate the acceleration for the fuel pod.

Dividing the net force of

4,800 \text{ N}

by the mass of

4,000 \text{ kg}

yields

1.2 \text{ m/s}^2

.

Applying Newton's Second Law directly.

Key Concept

Newton's Second Law (

F = ma

)

Estimated Time:1m 30s

Question 2033Question →

A hydraulic press ram must reach a downward operating velocity of $0.8 \text{ m/s}$ starting from rest. If the hydraulic system is calibrated to accelerate the ram at a constant rate of $0.2 \text{ m/s}^2$ , how many seconds will it take for the ram to reach its target velocity?

0.16 \text{ s}

0.25 \text{ s}

4.0 \text{ s}

40.0 \text{ s}

Show answer & explanation

Answer: $4.0 \text{ s}$

Answer

It will take

4.0 \text{ s}

for the hydraulic press ram to reach the target velocity.

The time required to reach a specific velocity from rest is calculated by dividing the target velocity by the constant acceleration (

t = \frac{v}{a}

). Dividing

0.8 \text{ m/s}

by

0.2 \text{ m/s}^2

results exactly in

4.0 \text{ seconds}

.

Step-by-Step Solution

1

Identify the known kinematic variables from the scenario.

Initial velocity is

0 \text{ m/s}

, final velocity is

0.8 \text{ m/s}

, and acceleration is

0.2 \text{ m/s}^2

.

Organizing the known values determines which rate formula must be applied.

2

Rearrange the acceleration formula to solve for time.

The formula

a = \frac{\Delta v}{t}

becomes

t = \frac{\Delta v}{a}

.

To isolate the unknown variable (time) from the standard acceleration equation.

3

Calculate the final elapsed time.

0.8 \div 0.2 = 4.0 \text{ seconds}

.

Executing the division yields the time required to reach the target velocity.

Key Concept

Kinematic Rate Equations (Solving for Time)

Estimated Time:45s

Question 2034Question →

A naval submarine uses internal ballast tanks to take in or pump out seawater, allowing it to change its overall weight while its total external volume remains exactly the same. Arrange the following operational states of the submarine in order of the submarine's OVERALL AVERAGE DENSITY, from LEAST dense to MOST dense.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from least dense to most dense is: Floating on the surface, followed by Hovering motionless, and finally Resting on the ocean floor.

Density directly dictates buoyancy states. An object with a density lower than seawater will float on the surface. An object with a density equal to seawater will hover neutrally. An object with a density greater than seawater will sink to the bottom. Therefore, the order from least to most dense is: floating, hovering, sinking.

Step-by-Step Solution

1

Analyze the condition required for an object to float.

The state of 'Floating on the surface' represents the lowest density.

According to Archimedes' principle, an object floats (positive buoyancy) when its overall density is less than the density of the fluid it is in.

2

Analyze the condition required for an object to hover underwater.

The state of 'Hovering motionless' represents a medium density.

An object achieves neutral buoyancy and hovers when its weight perfectly matches the buoyant force of the fully displaced fluid, meaning its density is exactly equal to the fluid's density.

3

Analyze the condition required for an object to sink.

The state of 'Resting on the solid ocean floor' represents the highest density.

An object sinks (negative buoyancy) when its weight exceeds the maximum possible buoyant force, meaning its overall density is greater than the fluid's density.

Key Concept

Buoyancy and Density Relationships (Positive, Neutral, and Negative Buoyancy)

Estimated Time:45s

Question 2035Question →

A hand-cranked emergency power generator uses a two-stage compound gear train to increase the rotational speed of its rotor. The hand crank is connected directly to the driving gear, Gear A, which has $60$ teeth. Gear A meshes with Gear B, which has $15$ teeth. Gear B shares a solid intermediate shaft with Gear C, which has $40$ teeth. Finally, Gear C meshes with Gear D on the generator's rotor shaft, which has $10$ teeth.

If a user turns the hand crank at a steady rate of $30\text{ rpm}$ , what is the resulting rotational speed of the generator rotor in $\text{rpm}$ ?

Show answer & explanation

Answer: 480

Answer

480

The correct answer is 480 because the total speed multiplication of a compound gear train is the product of the individual gear ratios of each stage. Stage 1 has a ratio of 60:15 (a 4:1 multiplier). Stage 2 has a ratio of 40:10 (a 4:1 multiplier). The total multiplier is 4 * 4 = 16. The input speed of 30 rpm multiplied by 16 gives a final output speed of 480 rpm.

Step-by-Step Solution

1

Calculate the speed multiplication factor of the first gear pair.

Factor of 4

Gear A (60 teeth) drives Gear B (15 teeth). The ratio of teeth is 60/15 = 4. Because a larger gear is driving a smaller gear, the rotational speed increases by this factor.

2

Calculate the rotational speed of the intermediate shaft.

120 rpm

Multiply the input speed (30 rpm) by the first stage factor (4). 30 * 4 = 120 rpm. Both Gear B and Gear C spin at this exact rate since they are mounted on the same solid shaft.

3

Calculate the speed multiplication factor of the second gear pair.

Factor of 4

Gear C (40 teeth) drives Gear D (10 teeth). The ratio is 40/10 = 4. Again, a larger gear is driving a smaller one, increasing the speed.

4

Calculate the final rotational speed of the rotor.

480 rpm

Multiply the intermediate shaft speed (120 rpm) by the second stage factor (4). 120 * 4 = 480 rpm.

Key Concept

Calculating output speed in a compound gear train

Question 2036Question →

A technician must remove a stuck brass bushing from inside a solid steel housing. The coefficient of thermal expansion for brass is greater than that of steel, meaning brass expands and contracts more for any given temperature change. Which of the following thermal procedures would be most effective to loosen the tight fit?

Cool the entire assembly uniformly

Heat the entire assembly uniformly

Apply heat directly to the brass bushing

Apply a cooling agent exclusively to the steel housing

Show answer & explanation

Answer: Cool the entire assembly uniformly

Answer

Cool the entire assembly uniformly.

Because the brass bushing has a higher coefficient of thermal contraction, it will shrink at a faster rate than the steel housing when both are cooled. As the temperature drops, the outer diameter of the brass bushing decreases more significantly than the inner diameter of the steel hole, which opens up clearance between the two parts and loosens the fit.

Step-by-Step Solution

1

Analyze the mechanical relationship between the components.

The brass bushing is the inner component, and the steel housing is the outer component. To loosen the fit, the outer diameter of the inner component must become smaller relative to the inner diameter of the outer component.

Identifying the relative positions of the parts dictates whether expansion or contraction is needed to create clearance.

2

Evaluate the effects of uniform temperature changes based on material properties.

Since brass has a higher coefficient of thermal expansion than steel, a uniform temperature increase will cause the brass to expand more than the steel. A uniform temperature decrease will cause the brass to contract more than the steel.

Different materials change size at different rates when subjected to the same temperature variations.

3

Determine which thermal treatment creates the desired clearance.

Heating uniformly makes the inner brass part expand more than the outer steel hole, tightening the fit. Cooling uniformly makes the inner brass part shrink more than the outer steel hole, loosening the fit.

Selecting the condition that reduces the inner part's size more rapidly than the outer part's size provides the mechanical clearance needed for removal.

Key Concept

Differential Thermal Expansion and Contraction

Question 2037Question →

A factory conveyor belt transports manufactured parts a distance of $45 \text{ meters}$ across the production floor in exactly $15 \text{ seconds}$ . What is the average speed of the conveyor belt in meters per second?

Show answer & explanation

Answer: 3

Answer

The average speed of the conveyor belt is

3 \text{ m/s}

.

The average speed is correctly calculated by dividing the total distance traveled (

45 \text{ meters}

) by the total time taken (

15 \text{ seconds}

), which equals

3 \text{ m/s}

.

Step-by-Step Solution

1

Identify the given values for distance and time.

Distance (

d

) =

45 \text{ meters}

, Time (

t

) =

15 \text{ seconds}

.

These variables are required to calculate average speed.

2

Apply the average speed formula.

v = \frac{d}{t} = \frac{45}{15}

.

Average speed is defined as the total distance traveled divided by the elapsed time.

3

Calculate the final speed.

v = 3

.

To find the numerical value of the speed in meters per second.

Key Concept

Average Speed

Question 2038Question →

During a facility upgrade, an industrial overhead crane is lifting a large generator component vertically. The component has a mass of $2,500 \text{ kg}$ . The crane's hoist mechanism applies an upward tension force of $35,000 \text{ N}$ . Assuming the acceleration due to gravity is $10 \text{ m/s}^2$ , what is the upward acceleration of the generator component?

0.25 \text{ m/s}^2

4.0 \text{ m/s}^2

12.8 \text{ m/s}^2

14.0 \text{ m/s}^2

Show answer & explanation

Answer: $4.0 \text{ m/s}^2$

Answer

The upward acceleration of the generator component is

4.0 \text{ m/s}^2

.

To find the acceleration, we first determine the net force acting on the component. The crane pulls up with

35,000 \text{ N}

, but gravity pulls down with a force equal to the object's weight. The weight is the mass times gravity (

2,500 \text{ kg} \times 10 \text{ m/s}^2 = 25,000 \text{ N}

). The net upward force is the difference between these two opposing forces (

35,000 \text{ N} - 25,000 \text{ N} = 10,000 \text{ N}

). Finally, applying Newton's Second Law (

a = F_{net} / m

), we divide the net force by the mass (

10,000 \text{ N} / 2,500 \text{ kg}

) to find an acceleration of

4.0 \text{ m/s}^2

.

Step-by-Step Solution

1

Calculate the downward weight force of the generator component.

Weight (

W

) =

m \times g

=

2,500 \text{ kg} \times 10 \text{ m/s}^2

=

25,000 \text{ N}

.

Gravity pulls down on the mass, creating an opposing force that must be overcome to lift the component.

2

Determine the net force acting on the component.

Net Force (

F_{net}

) = Upward Force - Downward Weight =

35,000 \text{ N} - 25,000 \text{ N}

=

10,000 \text{ N}

.

Newton's Second Law requires the total net force, taking all acting forces (tension and gravity) into account.

3

Calculate the acceleration using Newton's Second Law (

F = ma

).

Acceleration (

a

) =

F_{net} / m

=

10,000 \text{ N} / 2,500 \text{ kg}

=

4.0 \text{ m/s}^2

.

Acceleration is directly proportional to the net force and inversely proportional to the mass.

Key Concept

Newton's Second Law requires determining the net force by summing all forces acting on an object (including gravity) before dividing by mass to find acceleration.

Question 2039Question →

A uniform window-washing scaffold is $20$ feet long and weighs $180$ pounds. It is supported horizontally by two vertical cables, one attached at the extreme left end and one at the extreme right end. A worker weighing $220$ pounds stands exactly $5$ feet from the left cable. If the system is in static equilibrium, what is the tension, in pounds, in the right cable?

Show answer & explanation

Answer: 145

Answer

145

The system is in equilibrium, so the sum of the moments about the left support must be zero. The downward weights create a clockwise moment of

2900 \text{ lb-ft}

(

180 \text{ lbs} \times 10 \text{ ft}

plus

220 \text{ lbs} \times 5 \text{ ft}

). To balance this, the right cable, located

20

feet away, must provide an equal counter-clockwise moment. Dividing

2900

by

20

yields a tension of

145

pounds.

Step-by-Step Solution

1

Select a pivot point to calculate moments (torques).

The left end of the scaffold is chosen as the pivot point.

Choosing the left end eliminates the unknown tension of the left cable from the equation, allowing direct calculation of the right cable's tension.

2

Calculate the total clockwise moment caused by the downward weights.

Scaffold moment =

180 \text{ lbs} \times 10 \text{ ft} = 1800 \text{ lb-ft}

. Worker moment =

220 \text{ lbs} \times 5 \text{ ft} = 1100 \text{ lb-ft}

. Total clockwise moment =

2900 \text{ lb-ft}

.

The scaffold's weight acts at its center (

10

ft), and the worker's weight acts at

5

ft from the pivot. Moment equals force multiplied by distance.

3

Equate the clockwise moment to the counter-clockwise moment and solve.

T_R \times 20 \text{ ft} = 2900 \text{ lb-ft}

, which gives

T_R = 145 \text{ lbs}

.

For the scaffold to be in static equilibrium, the sum of all moments about any pivot point must be zero.

Key Concept

Rotational equilibrium and calculation of moments (torque) to find support reaction forces.

Question 2040Question →

A laboratory container is filled to the very brim with water. A 3-kilogram solid wood block is gently placed into the water, where it floats, and the water that spills over is collected and weighed. The wood is then removed, the container is completely refilled, and a 3-kilogram solid steel weight is lowered into the water until it rests on the bottom. The water that spills over from the steel is also collected and weighed. Based on Archimedes' principle, which of the following statements correctly compares the weight of the water displaced by the two objects?

The wood block displaces a greater weight of water than the steel weight.

The steel weight displaces a greater weight of water because it completely submerges beneath the surface.

Both objects displace the exact same weight of water because their initial masses are identical.

The steel weight displaces a greater weight of water because steel has a higher density than wood.

Show answer & explanation

Answer: The wood block displaces a greater weight of water than the steel weight.

Answer

The wood block displaces a greater weight of water than the steel weight.

According to Archimedes' principle, a floating object (like the wood) displaces a weight of fluid equal to its own weight, meaning the wood displaces 3 kg of water. A sinking object (like the steel) displaces a volume of fluid equal to its own volume. Since steel is very dense, a 3 kg piece of steel takes up very little volume and therefore displaces much less than 3 kg of water. Thus, the floating wood displaces more water than the sinking steel.

Step-by-Step Solution

1

Analyze the displaced water for the floating wood block.

According to Archimedes' principle, a floating object displaces a weight of fluid equal to its own weight. Therefore, the 3-kilogram wood block displaces exactly 3 kilograms of water.

This establishes the buoyant force required to keep the wood in equilibrium on the surface of the water.

2

Analyze the displaced water for the sinking steel weight.

A sinking object displaces a volume of fluid equal to its own total volume. Because steel is much denser than water, a 3-kilogram piece of steel has a very small volume.

This determines how much water is physically pushed out of the container by the fully submerged steel.

3

Compare the weight of the water displaced by both objects.

The wood displaces 3 kilograms of water. The steel displaces a volume of water equal to its own small volume, which weighs much less than 3 kilograms. Therefore, the floating wood displaces a greater weight of water.

Directly answering the question by comparing the calculated fluid displacements.

Key Concept

Buoyancy and Archimedes' Principle

Answer

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