All practice questions

Question 1861Question →

A compact utility tractor applies a constant net forward force of $1,200 \text{ N}$ to push a stalled passenger vehicle on a level, paved road. If the vehicle accelerates at a rate of $1.5 \text{ m/s}^2$ , what is the mass of the vehicle?

Show answer & explanation

Answer: 800

Answer

800 kg

According to Newton's Second Law (

F=ma

), the mass of an object is equal to the net force acting upon it divided by its acceleration. By dividing the applied net force of

1,200 \text{ N}

by the acceleration of

1.5 \text{ m/s}^2

, we determine the mass of the vehicle is

800 \text{ kg}

.

Step-by-Step Solution

1

Identify the known variables provided in the problem.

Net Force (

F

) =

1,200 \text{ N}

, Acceleration (

a

) =

1.5 \text{ m/s}^2

.

Extracting the given information is the first step to determine which physics formula to apply.

2

State the formula for Newton's Second Law of Motion.

F = ma

This formula establishes the mathematical relationship between net force, mass, and acceleration.

3

Rearrange the formula to solve for the unknown variable, mass (

m

).

m = \frac{F}{a}

To find the mass, it must be isolated algebraically on one side of the equation.

4

Substitute the known values into the rearranged formula and compute the result.

m = \frac{1,200}{1.5} = 800

Executing the division calculation provides the final mass in kilograms.

Key Concept

Newton's Second Law of Motion (F=ma)

Question 1862Question →

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

Click a left item, then click its matching right item

Show answer & explanation

Answer

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.

Step-by-Step Solution

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.

Key Concept

Mechanical Advantage and Function of Gear Trains

Question 1863Question →

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

Click a left item, then click its matching right item

Show answer & explanation

Answer

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).

Step-by-Step Solution

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.

Key Concept

Ideal Mechanical Advantage (IMA) of a Wheel and Axle

Question 1864Question →

A mechanical design requires a structural component to act as a drive shaft, meaning it will primarily experience twisting forces (torque) along its longitudinal axis. Evaluate this statement: A closed hollow circular tube provides greater torsional stiffness and resistance to twisting than an open structural shape, such as an I-beam or C-channel, of equal weight and material.

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is accurate. For a given weight of material, a closed hollow circular profile is the most efficient cross-sectional shape for resisting torsional (twisting) loads because it distributes material in a continuous loop along the outer edge. Open profiles like I-beams or C-channels are designed specifically for bending moments and are highly susceptible to warping under torsion.

Step-by-Step Solution

1

Identify the primary force acting on the structural component based on the given scenario.

The component acts as a drive shaft, which means it experiences twisting forces, known as torsion or torque.

Different structural shapes are optimized to resist specific types of mechanical forces (e.g., bending, compression, torsion).

2

Analyze how different cross-sectional profiles handle torsional loads.

Closed profiles (like circular tubes) resist torsion efficiently, while open profiles (like I-beams or C-channels) have low torsional stiffness.

Torsional stress is best resisted when the material forms a continuous, unbroken loop far from the central axis, maximizing the polar moment of inertia.

3

Evaluate the accuracy of the statement based on the mechanical analysis.

The statement correctly claims that the closed hollow circular tube provides greater torsional stiffness than an open shape of equal weight.

This aligns with the fundamental mechanical principles governing structural profiles under torsion.

Key Concept

Torsional strength of closed versus open structural profiles

Question 1865Question →

A $2,000 \text{ kg}$ pickup truck is coasting along a level road at a velocity of $15 \text{ m/s}$ . What is the magnitude of the truck's momentum?

Show answer & explanation

Answer: 30000

Answer

30000

Momentum is directly proportional to both mass and velocity. By multiplying the

2,000 \text{ kg}

mass by the

15 \text{ m/s}

velocity, we determine the total momentum is

30,000 \text{ kg} \cdot \text{m/s}

.

Step-by-Step Solution

1

Identify the known variables from the text.

Mass (

m

) is

2,000 \text{ kg}

and velocity (

v

) is

15 \text{ m/s}

.

Momentum calculation requires both the mass and the velocity of the object.

2

Apply the formula for momentum.

p = m \times v

The linear momentum of an object is defined as the product of its mass and its velocity.

3

Calculate the final value.

p = 2,000 \times 15 = 30,000

Multiplying the mass by the velocity yields the total magnitude of momentum.

Key Concept

Momentum is the product of an object's mass and its velocity (

p = mv

).

Question 1866Question →

In a vacuum testing chamber, two identical engine heat sinks—one with a highly polished aluminum finish and the other painted matte black—are heated to exactly 200°C. True or false: The matte black heat sink will emit thermal radiation at a significantly higher rate than the polished aluminum one.

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is true because a surface's ability to emit thermal radiation (its emissivity) heavily depends on its color and texture. Dark, matte surfaces are almost ideal emitters of thermal radiation, whereas shiny, polished metals are highly reflective and trap heat internally, making them very poor emitters. Consequently, the matte black heat sink radiates its thermal energy into the vacuum much faster than the polished one.

Step-by-Step Solution

1

Identify the available mode of heat transfer in the described environment.

Because the heat sinks are in a vacuum, thermal energy can only be transferred via radiation. Conduction and convection require a physical medium (like air or liquid) and are not possible here.

Establishing the correct heat transfer mechanism is critical to prevent applying the wrong physical rules.

2

Evaluate the effect of surface finish on thermal radiation emission.

A matte black finish has high emissivity, meaning it is an excellent radiator of thermal energy. A highly polished aluminum finish has low emissivity, making it a very poor radiator.

Emissivity is the key material property that dictates how efficiently a surface emits thermal radiation.

3

Compare the heat emission rates of the two heat sinks.

Since both heat sinks are at the exact same temperature, the matte black heat sink will emit radiant heat at a significantly higher rate than the polished aluminum one.

At any given temperature, an object with a higher emissivity always radiates more thermal power than an object with lower emissivity.

Key Concept

Surface Emissivity and Thermal Radiation

Estimated Time:45s

Question 1867Question →

Match each specialized fastener or component to the primary mechanical function it serves in an assembly.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Shoulder Bolt matches with providing a bearing surface; Belleville Washer matches with compensating for thermal expansion; Castellated Nut matches with physically blocking rotation; Blind Rivet matches with expanding against an inaccessible panel.

Each fastener has a specific design intended for a unique mechanical application. Shoulder bolts provide bearing surfaces, Belleville washers maintain tension through spring action, castellated nuts provide positive locking via a pin, and blind rivets allow for one-sided installation.

Step-by-Step Solution

1

Analyze the function of a Shoulder Bolt.

Identified that it has an unthreaded 'shoulder' section.

This unthreaded section acts as a dowel or pivot point for linkages, pulleys, or sliding components.

2

Analyze the function of a Belleville Washer.

Identified that it is a conical spring washer.

Its conical shape allows it to act as a heavy spring, absorbing vibrations and thermal expansion to maintain bolt tension.

3

Analyze the function of a Castellated Nut.

Identified that it has slots cut into one end.

These slots are designed to receive a cotter pin that passes through the bolt, creating a positive lock against rotation.

4

Analyze the function of a Blind Rivet.

Identified that it is used when only one side of the workpiece is accessible.

A tool pulls a mandrel that deforms the blind end, clamping the materials together before the mandrel breaks off.

Key Concept

Specialized Fasteners and Locking Mechanisms

Question 1868Question →

A technician places two identical steel tanks outdoors in direct sunlight. Tank X is painted flat black, while Tank Y is painted glossy white. After a few hours, Tank X is significantly hotter to the touch than Tank Y. What is the primary reason for this temperature difference?

Flat black surfaces are better absorbers of thermal radiation than glossy white surfaces.

Glossy white surfaces restrict the thermal convection of the surrounding air.

Flat black paint increases the specific heat capacity of the steel tank.

Glossy white paint prevents the direct conduction of heat from the sun.

Show answer & explanation

Answer: Flat black surfaces are better absorbers of thermal radiation than glossy white surfaces.

Answer

Flat black surfaces are better absorbers of thermal radiation than glossy white surfaces.

The correct answer accurately states that flat black surfaces are better absorbers of thermal radiation. Heat from the sun travels as electromagnetic radiation. Dark, matte finishes absorb most of this radiant energy and convert it into heat, whereas light, glossy finishes reflect most of the radiant energy away.

Step-by-Step Solution

1

Identify the source and method of heat transfer.

The sun heats the tanks through the vacuum of space, meaning the heat transfer method must be thermal radiation.

Conduction requires direct contact, and convection requires a fluid medium, leaving radiation as the only way solar energy reaches the tanks.

2

Analyze how the different surface finishes interact with thermal radiation.

The flat black tank (Tank X) absorbs more radiant energy, while the glossy white tank (Tank Y) reflects more of it.

Dark, matte surfaces are excellent absorbers of electromagnetic radiation, whereas light, shiny surfaces are excellent reflectors.

Key Concept

Thermal radiation absorption depends heavily on an object's surface color and texture.

Question 1869Question →

A solid steel engine block with a true weight of $2,000 \text{ N}$ is completely submerged in a freshwater lake. The block displaces a volume of water that weighs $250 \text{ N}$ . According to Archimedes' principle, what is the magnitude of the upward buoyant force acting on the engine block?

2,000 \text{ N}

1,750 \text{ N}

2,250 \text{ N}

250 \text{ N}

Show answer & explanation

Answer: $250 \text{ N}$

Answer

250 \text{ N}

According to Archimedes' principle, the upward buoyant force on an object submerged in a fluid is exactly equal to the weight of the fluid it displaces. Since the engine block displaces water weighing

250 \text{ N}

, the buoyant force acting upon it is exactly

250 \text{ N}

.

Step-by-Step Solution

1

Identify the values provided in the scenario.

The true weight of the block is

2,000 \text{ N}

and the weight of the displaced water is

250 \text{ N}

.

Extracting given values clarifies which numbers represent which physical properties.

2

Apply Archimedes' principle to determine the buoyant force.

Archimedes' principle states that the upward buoyant force is equal to the weight of the displaced fluid.

This principle directly links the volume of fluid displaced to the upward force exerted by the fluid.

3

Select the value that represents the weight of the displaced fluid.

The weight of the displaced water is

250 \text{ N}

, therefore the buoyant force is

250 \text{ N}

.

The object's own weight does not determine the buoyant force; only the displaced fluid's weight matters.

Key Concept

Archimedes' Principle and Buoyant Force

Question 1870Question →

A $120$ -pound crate rests on a sturdy $12$ -foot plank spanning two sawhorses placed at opposite ends. The center of the crate is located $3$ feet from the left sawhorse. Assuming the weight of the plank is negligible, what is the downward force, in pounds, exerted by the crate on the left sawhorse?

Show answer & explanation

Answer: 90

Answer

The downward force exerted on the left sawhorse is

90

pounds.

Because the crate is positioned

3

feet from the left sawhorse and

9

feet from the right sawhorse on a

12

-foot span, the left sawhorse must support

\frac{9}{12}

(or

75\%

) of the crate's weight. Multiplying

120

pounds by

0.75

yields exactly

90

pounds.

Step-by-Step Solution

1

Calculate the distance from the crate to the right sawhorse.

12 - 3 = 9

feet.

The weight supported by one end of a beam is proportional to the distance from the load to the opposite end.

2

Determine the fraction of the total weight carried by the left sawhorse.

\frac{9}{12}

, which simplifies to

\frac{3}{4}

or

0.75

.

This fraction represents the ratio of the opposite distance to the total span, dictating how much load is distributed to the left support.

3

Calculate the downward force on the left sawhorse.

120 \times 0.75 = 90

pounds.

Multiplying the total crate weight by the calculated proportion gives the specific load carried by that support.

Key Concept

Load Distribution and Balancing

Estimated Time:45s

Question 1871Question →

Based on Newton's First Law (Inertia), match each physical scenario on the left with the correct description of its resulting motion or force balance on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The elevator scenario pairs with passengers continuing downward; the turning vehicle pairs with objects moving in a straight path (appearing to move right); the stationary block pairs with remaining stationary due to no unbalanced forces; and the skydiver pairs with balanced forces resulting in zero net force.

Each pairing correctly applies the principle of inertia to a specific physical system. Newton's First Law explains both why moving objects resist stopping or turning (the elevator and vehicle scenarios) and why objects maintain their current state when forces are balanced (the stationary block and the constant-velocity skydiver).

Step-by-Step Solution

1

Analyze the descending elevator scenario.

Matches with passengers feeling pushed downward.

Newton's First Law states objects in motion stay in motion. The elevator stops, but the passengers' inertia carries them downward momentarily.

2

Analyze the vehicle turning left.

Matches with unsecured objects sliding to the right.

Objects tend to travel in a straight line. As the car turns left underneath them, their straight-line inertia makes them move right relative to the car.

3

Analyze the stationary block on a frictionless table.

Matches with remaining completely stationary.

An object at rest stays at rest unless acted upon by an external force, regardless of the lack of friction.

4

Analyze the skydiver at terminal velocity.

Matches with downward gravity balanced by upward air resistance.

Terminal velocity is a constant speed. A constant speed in a straight line means acceleration is zero, requiring a net force of zero.

Key Concept

Newton's First Law dictates that an object's state of motion (either at rest or moving at a constant velocity) will not change unless an unbalanced external force acts upon it. This resistance to change is known as inertia.

Question 1872Question →

An aviation mechanic pushes a loaded tool cart across a level hangar floor. If the mechanic applies a net forward force of $120 \text{ N}$ and the cart accelerates at $2 \text{ m/s}^2$ , what is the mass of the cart?

60 \text{ kg}

240 \text{ kg}

12 \text{ kg}

120 \text{ kg}

Show answer & explanation

Answer: $60 \text{ kg}$

Answer

The mass of the tool cart is

60 \text{ kg}

.

According to Newton's Second Law (

F = ma

), mass is equal to net force divided by acceleration. Dividing

120 \text{ N}

by

2 \text{ m/s}^2

results in

60 \text{ kg}

.

Step-by-Step Solution

1

Identify the given variables from the scenario.

Net force (

F

) =

120 \text{ N}

, Acceleration (

a

) =

2 \text{ m/s}^2

.

These are the known values required to solve for the missing variable.

2

Apply Newton's Second Law of Motion.

The formula is

F = ma

.

This law defines the physical relationship between net force, mass, and acceleration.

3

Rearrange the formula to solve for mass (

m

).

m = \frac{F}{a}

To find the unknown mass, it must be isolated on one side of the equation.

4

Substitute the values and calculate the final result.

m = \frac{120}{2} = 60 \text{ kg}

Performing the division yields the actual mass of the object.

Key Concept

Newton's Second Law states that force equals mass times acceleration (

F = ma

), meaning an object's mass can be calculated by dividing the net force by its acceleration.

Question 1873Question →

In a mechanical flight control linkage, a solid steel pin is used to connect a single control rod (tang) inside a U-shaped bracket (clevis). A heavy tensile force is applied to the control rod, pulling it directly away from the fixed bracket. Which of the following best describes the primary type of stress experienced by the pin and how its load-bearing capacity compares to a simple two-plate lap joint secured by a single identical pin?

The pin experiences double shear stress, distributing the load across two cross-sectional areas and making it significantly stronger than a lap joint.

The pin experiences single shear stress, concentrating the load at one cross-sectional area and providing the same strength as a lap joint.

The pin experiences pure tensile stress because the control rod is being pulled, making it more susceptible to stretching than a lap joint.

The pin experiences compressive stress as the sides of the bracket squeeze inward, providing greater rigidity than a lap joint.

Show answer & explanation

Answer: The pin experiences double shear stress, distributing the load across two cross-sectional areas and making it significantly stronger than a lap joint.

Answer

The option stating the pin experiences double shear stress, distributing the load across two cross-sectional areas and making it significantly stronger than a lap joint.

The pulling force on the control rod acts perpendicular to the longitudinal axis of the connecting pin, which creates shear stress (a slicing force). Because the U-shaped clevis bracket supports the pin at two distinct points—above and below the control rod—the pin must be sheared in two separate places for the joint to fail. This configuration is known as 'double shear.' Because the load is distributed across two cross-sectional areas of the pin instead of just one, a double shear joint is significantly stronger than a simple lap joint, which only puts the fastener in single shear.

Step-by-Step Solution

1

Analyze the direction of the applied force relative to the pin.

The control rod is pulled horizontally, which applies a force perpendicular to the longitudinal axis of the vertical connecting pin.

Forces acting perpendicular to the axis of a fastener create shear stress, not tensile stress.

2

Examine the physical structure of the joint.

The joint consists of three overlapping sections: the upper arm of the clevis bracket, the central control rod (tang), and the lower arm of the clevis bracket.

Identifying the number of overlapping components determines how many places the pin could be severed.

3

Determine the number of shear planes.

Because the control rod pulls in the middle and the bracket resists on both the top and bottom, the pin would have to be sliced in two separate places to fail.

Two shear planes indicate 'double shear', which distributes the force over twice the cross-sectional area of the pin compared to a standard two-plate lap joint (single shear).

Key Concept

Identifying load distribution and shear stress (single vs. double shear) in mechanical joints.

Question 1874Question →

A $4,000 \text{ kg}$ empty railcar is rolling along a straight, level track at a velocity of $6 \text{ m/s}$ . It collides and automatically couples with a stationary loaded railcar that has a mass of $8,000 \text{ kg}$ . Assuming no external friction is acting on the railcars, what is the final velocity of the two coupled railcars immediately after the impact?

2 \text{ m/s}

3 \text{ m/s}

6 \text{ m/s}

19.6 \text{ m/s}

Show answer & explanation

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

Answer

The final velocity is

2 \text{ m/s}

.

By the law of conservation of momentum, the total momentum before the collision equals the total momentum after. The initial momentum is

4,000 \text{ kg} \times 6 \text{ m/s} = 24,000 \text{ kg}\cdot\text{m/s}

. After the railcars couple, the new total mass is

12,000 \text{ kg}

. Dividing the total momentum by the total mass gives a final velocity of

2 \text{ m/s}

.

Step-by-Step Solution

1

Calculate the total initial momentum of the system.

p_{initial} = 4,000 \text{ kg} \times 6 \text{ m/s} = 24,000 \text{ kg}\cdot\text{m/s}

Because momentum is conserved in the collision, finding the total momentum before impact is required.

2

Determine the total mass of the coupled railcars.

m_{total} = 4,000 \text{ kg} + 8,000 \text{ kg} = 12,000 \text{ kg}

In an inelastic collision where objects couple together, they move as a single combined mass after impact.

3

Calculate the final velocity using the conserved momentum and total mass.

v_{final} = \frac{24,000 \text{ kg}\cdot\text{m/s}}{12,000 \text{ kg}} = 2 \text{ m/s}

Dividing the total conserved momentum by the new total mass isolates the final velocity of the combined system.

Key Concept

Conservation of Momentum in Inelastic Collisions

Question 1875Question →

An engineering team is analyzing how different mechanical interactions affect the control and efficiency of an off-road vehicle. Match each vehicle component or scenario to the specific type of frictional resistance it primarily demonstrates.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: stationary tires demonstrate Static Friction; brake pads demonstrate Kinetic Friction; ball bearings demonstrate Rolling Friction; and moving through mud demonstrates Fluid Friction.

Each mechanical scenario perfectly isolates one of the four primary types of friction. Stationary objects resisting motion rely on static friction. Surfaces sliding past one another generate kinetic friction. Round objects moving over a surface experience rolling friction. Objects moving through a liquid or gas encounter fluid friction.

Step-by-Step Solution

1

Analyze the tire scenario

The tires are stationary relative to the rock surface.

Identifying that there is no relative motion points to static friction.

2

Analyze the brake pad scenario

The pads rub directly against the moving steel rotors.

Solid surfaces sliding against each other exhibit kinetic (sliding) friction.

3

Analyze the ball bearing scenario

The bearings roll along their inner and outer races.

Objects rolling over a surface experience rolling friction, which is much lower than sliding friction.

4

Analyze the mud scenario

The vehicle body pushes through a semi-liquid medium.

Moving through liquids or gases involves fluid (viscous) friction.

Key Concept

Types of Frictional Resistance in Mechanics

Question 1876Question →

An industrial processing facility uses two large vertical storage tanks that are open to the atmosphere. Tank A contains two distinct layers of non-mixing liquids: a $5\text{-meter}$ top layer of oil (density = $800\text{ kg/m}^3$ ) and a $3\text{-meter}$ bottom layer of water (density = $1000\text{ kg/m}^3$ ). Tank B, which has exactly twice the cross-sectional area of Tank A, contains pure water to a total depth of $8\text{ meters}$ . Which of the following statements correctly compares the gauge hydrostatic pressure acting on the interior bottom surfaces of the two tanks?

The pressure at the bottom of Tank B is greater primarily because its larger cross-sectional area holds a much larger total volume and weight of fluid.

The pressures are exactly equal because the total depth of the fluid column is

8\text{ meters}

in both tanks, regardless of the container's base area.

The pressure at the bottom of Tank B is greater because the entire

8\text{-meter}

fluid column consists of the denser liquid.

The pressure at the bottom of Tank A is greater because the oil layer physically compresses the water layer below it, multiplying the downward force.

Show answer & explanation

Answer: The pressure at the bottom of Tank B is greater because the entire $8\text{-meter}$ fluid column consists of the denser liquid.

Answer

The pressure at the bottom of Tank B is greater because its entire fluid column is composed of the denser water, yielding a higher hydrostatic pressure than Tank A's mixed column.

The correct answer accurately calculates that Tank B has a higher bottom pressure because it contains a full

8\text{-meter}

column of the denser water (

1000\text{ kg/m}^3

), while Tank A replaces

5\text{ meters}

of that column with lighter oil (

800\text{ kg/m}^3

). It also correctly implies that hydrostatic pressure depends only on fluid density and depth, not on the tank's cross-sectional area.

Step-by-Step Solution

1

Identify the formula for gauge hydrostatic pressure.

The hydrostatic pressure

P

at a depth

h

is calculated using

P = \rho gh

, where

\rho

is the fluid density and

g

is the acceleration due to gravity.

This formula is needed to find the pressure at the bottom of each tank.

2

Calculate the relative pressure at the bottom of Tank A.

For multiple layers, sum the individual pressures:

P_A = (\rho_{oil} \times g \times h_{oil}) + (\rho_{water} \times g \times h_{water}) = (800 \times g \times 5) + (1000 \times g \times 3) = 4000g + 3000g = 7000g\text{ Pa}

.

The total bottom pressure in a multi-fluid system is the combined pressure of all stacked fluid layers.

3

Calculate the relative pressure at the bottom of Tank B.

Tank B contains only water:

P_B = \rho_{water} \times g \times h_{total} = 1000 \times g \times 8 = 8000g\text{ Pa}

.

This calculation provides the value needed to compare Tank B against Tank A.

4

Compare the pressures and analyze the effect of tank width.

Since

8000g > 7000g

, Tank B has the higher bottom pressure. The larger cross-sectional area of Tank B is irrelevant to the calculation.

Hydrostatic pressure depends solely on depth and density, meaning a wider container does not artificially increase pressure.

Key Concept

Hydrostatic pressure depends strictly on fluid density, gravity, and depth (

P = \rho gh

), and is independent of the container's shape, base area, or total volume of fluid.

Question 1877Question →

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.

Show answer & explanation

Answer: It will rotate slower and produce more torque.

Answer

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.

Step-by-Step Solution

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.

Key Concept

Mechanical Advantage in Gears

Question 1878Question →

An industrial furnace utilizes a tall vertical exhaust stack to naturally vent out combustion gases without the use of mechanical fans. Arrange the following stages of this natural convection cycle in the correct chronological order, beginning immediately after the fuel is ignited.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The sequence begins with the transfer of thermal energy (heating), followed by volumetric expansion (density reduction), leading to upward movement (buoyancy), and concluding with cooler air being drawn in to replace the rising gases (draft creation).

Natural convection is entirely dependent on density differences. Thermal energy must first be applied to the fluid, causing it to expand. This volumetric expansion lowers the fluid's density, creating a buoyant force relative to the surrounding cooler fluid. The buoyant hot fluid rises, and its departure creates a localized pressure drop at the bottom, which pulls in cooler, denser fluid to continue the cycle.

Step-by-Step Solution

1

Identify the primary source of thermal energy.

Combustion heats the gases, increasing kinetic energy.

Convection must be driven by an initial temperature differential.

2

Determine the physical reaction of a fluid to increased heat.

The gases expand and their density decreases.

As molecules gain energy, they move further apart, occupying more volume for the same mass.

3

Apply the principles of fluid mechanics to fluids of differing densities.

The lighter, less dense hot gases rise.

Gravity pulls down harder on the denser, colder air, forcing the less dense, hotter air upward.

4

Trace the resulting effect of the moving fluid on the overall system.

Cooler air is pulled into the base of the stack.

The evacuated space at the base creates a relative vacuum, which is immediately filled by denser ambient air, sustaining the cycle.

Key Concept

Density-Driven Natural Convection (The Stack Effect)

Question 1879Question →

A barber chair operates using a closed hydraulic lifting system to raise a customer. The input pump piston has a cross-sectional area of $4 \text{ in}^2$ , and the output lifting piston has a cross-sectional area of $60 \text{ in}^2$ . If the barber pushes the input piston down a distance of $7.5 \text{ inches}$ , how many inches does the output piston move upward?

Show answer & explanation

Answer: 0.5

Answer

0.5

In a hydraulic system, the mechanical advantage that multiplies the output force must correspondingly divide the output distance. The output piston has an area 15 times larger than the input piston (

60 / 4 = 15

). Therefore, while it can lift 15 times the force, it will only travel

1/15\text{th}

of the input distance (

7.5 / 15 = 0.5 \text{ inches}

).

Step-by-Step Solution

1

Identify the relevant principle for hydraulic piston travel distances.

The volume of fluid displaced by the input piston equals the volume gained by the output piston:

A_1 \times d_1 = A_2 \times d_2

.

Because hydraulic fluid is essentially incompressible, the total volume in the closed system remains constant.

2

Set up the equation using the given values.

4 \times 7.5 = 60 \times d_2

Input area (

A_1

) is 4, input distance (

d_1

) is 7.5, and output area (

A_2

) is 60.

3

Calculate the total fluid volume displaced.

4 \times 7.5 = 30 \text{ in}^3

Multiplying area by distance gives the volume of fluid moved.

4

Solve for the output piston's upward distance (

d_2

).

30 / 60 = 0.5 \text{ inches}

Dividing the displaced volume by the output piston's larger area yields the smaller travel distance.

Key Concept

Conservation of volume in hydraulic systems dictates that distance and area are inversely proportional (Mechanical Advantage trades distance for force).

Question 1880Question →

A temperature sensor utilizes a bimetallic strip composed of a top layer of brass and a bottom layer of steel, which are permanently bonded together. Brass has a significantly higher coefficient of thermal expansion than steel. If the strip is perfectly straight at room temperature, what will happen to its physical shape when it is exposed to sub-zero temperatures?

It bends into a curve with the brass layer on the inside radius.

It bends into a curve with the steel layer on the inside radius.

It remains completely straight but decreases in overall length.

It remains completely straight and increases in overall length.

Show answer & explanation

Answer: It bends into a curve with the brass layer on the inside radius.

Answer

The strip bends into a curve with the brass layer on the inside radius.

When cooled, metals contract. Because brass has a higher coefficient of thermal expansion than steel, it contracts (shrinks) significantly more than steel for the same drop in temperature. Since the two layers are permanently bonded, the shorter brass layer pulls the strip into a curve, acting as the inner, shorter radius of that curve.

Step-by-Step Solution

1

Identify the temperature change.

The strip is exposed to sub-zero temperatures, meaning it is being cooled.

Cooling causes most metals to contract (shrink) rather than expand.

2

Compare the contraction rates of the two metals.

Because brass has a higher coefficient of thermal expansion than steel, it will contract more than steel when cooled.

The coefficient of thermal expansion dictates the rate of both expansion during heating and contraction during cooling.

3

Determine the physical effect on the bonded strip.

The brass layer becomes shorter than the steel layer. Because they are permanently bonded, the strip must curve to accommodate the different lengths.

Bonded materials with unequal lengths naturally form an arc to relieve internal stress.

4

Identify the inner and outer layers of the resulting curve.

The shorter layer (brass) forms the inside of the curve, while the longer layer (steel) forms the outside.

Geometrically, the inner radius of any curve has a shorter path length than its outer radius.

Key Concept

Differential Thermal Contraction in Bimetallic Strips

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