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A Makeshift Elevator. While exploring an elaborate tunnel system, you and your team get lost and find yourselves at the bottom of a \(450-\mathrm{m}\) vertical shaft. Suspended from a thick rope (near the floor) is a large rectangular bucket that looks like it had been used to transport tools and debris up and down the tunnel. Mounted on the floor near one of the walls is a gasoline engine \((3.5 \mathrm{hp})\) that turns a pulley and rope, and a sign that reads "Emergency Lift." It is clear that the engine is used to drive the bucket up the shaft. On the wall next to the engine is a sign indicating that a full tank of gas will last exactly 15 minutes when the engine is running at full power. You open the engine's gas tank and estimate that it is \(1 / 4\) full, and there are no other sources of gasoline. (a) Assuming zero friction, if you send your team's lightest member (who weighs \(125 \mathrm{lb}\) ), and the bucket weighs \(150 \mathrm{lb}\) when empty, how far up the shaft will the engine take her (and the bucket)? Will it get her out of the mine? (b) Assuming an effective collective friction (from the pulleys, etc.) of \(\mu_{\mathrm{eff}}=0.10\) (so that \(F_{\mathrm{f}}=\mu_{\mathrm{eff}} M g,\) where \(M\) is the total mass of the bucket plus team member), will the engine (with a \(1 / 4\) -full tank of gas) lift her to the top of the shaft?

Step by step solution

01

Convert Engine Power to Watts

First, we convert the power of the engine from horsepower to watts. We know that 1 horsepower is equivalent to 746 watts. Therefore, the engine's power in watts is given by:\[ P = 3.5 \text{ hp} \times 746 \text{ W/hp} = 2611 \text{ W} \]

02

Calculate the Work Done by the Engine

Next, we calculate the total work that can be done by the engine. The time that the engine can run with a full tank is 15 minutes, which is 900 seconds. Since the tank is \( \frac{1}{4} \) full, the duration is:\[ t = \frac{900}{4} = 225 \text{ seconds} \]The work done by the engine is power times time:\[ W = P \times t = 2611 \text{ W} \times 225 \text{ s} = 587,475 \text{ J} \]

03

Calculate the Total Weight and Potential Energy

The total weight (in pounds) is the weight of the team member plus the empty bucket:\[ W_{total} = 125 \text{ lb} + 150 \text{ lb} = 275 \text{ lb} \]Convert this to newtons using the conversion 1 lb = 4.448 N:\[ W_{total, N} = 275 \text{ lb} \times 4.448 \text{ N/lb} = 1223.2 \text{ N} \]

04

Determine the Distance Without Friction

Without friction, the work done goes entirely into lifting. Using the formula for potential energy \( U = mgh \), where \( h \) is the height:\[ W = W_{total, N} \times h \]Solving for \( h \) gives:\[ h = \frac{W}{W_{total, N}} = \frac{587,475 \text{ J}}{1223.2 \text{ N}} = 480.34 \text{ meters} \]The engine can lift the member and bucket out because 480.34 m > 450 m.

05

Calculate the Work Against Friction

With friction, there is additional work needed against friction, which is given by the effective friction force:\[ F_{f} = \mu_{eff} \times W_{total, N} = 0.10 \times 1223.2 \text{ N} = 122.32 \text{ N} \]The work against friction when travelling a distance \( h \) is:\[ W_{friction} = F_{f} \times h \]

06

Determine the Height With Friction

The total work done by the engine is used to lift the bucket and fight against friction. Thus we use:\[ W = W_{total, N} \times h + F_{f} \times h = h (W_{total, N} + F_{f}) \]Solving for \( h \):\[ h = \frac{W}{W_{total, N} + F_{f}} = \frac{587,475 \text{ J}}{1223.2 \text{ N} + 122.32 \text{ N}} = \frac{587,475}{1345.52} \approx 436.53 \text{ meters} \]The engine does not lift her to 450 m with friction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Friction

Friction is the force that resists the relative motion of surfaces sliding against each other. It plays a critical role in everyday experiences and engineering applications.
When considering the makeshift elevator scenario, friction comes from the interaction between the pulleys and the load.
This force is often represented as:

• Static friction (before motion starts)
• Kinetic friction (while surfaces slide)

For our calculation, we consider effective kinetic friction using the coefficient of friction given (\( \mu_{\text{eff}} = 0.10 \).
The formula for calculating frictional force is given by:\[ F_{f} = \mu_{\text{eff}} \times M \times g \]where

• \( M \) is the mass
• \( g \) is acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))

Frictional forces cause energy loss, affecting how far up the engine can lift the bucket and person. Understanding this concept is key to solving problems involving motion and energy.

Work and Energy

Work and energy are fundamental concepts in physics, closely related through the Work-Energy Principle. Work is defined as the energy transferred to or from an object via a force causing displacement.
In this problem, the engine performs work to lift the bucket and person against gravitational force.
To calculate work done, use:\[ W = P \times t \]where

• \( W \) is work in joules
• \( P \) is power in watts
• \( t \) is time in seconds

The total work available is determined by the engine's power and remaining fuel.
This energy is divided into lifting the load and overcoming friction. By understanding these interactions, one can predict the machine's capability to perform tasks under varying conditions.

Power Conversion

Power is the rate of doing work or transferring energy. It's crucial for determining the engine's capacity to lift the bucket.
First, we converted the engine's power from horsepower to watts. This is essential because:

• Standard units ease calculations
• Watts (W) is the unit used for energy metrics

The conversion is done using the formula:\[ P = 3.5 \, \text{hp} \times 746 \, \text{W/hp} \approx 2611 \, \text{W} \]With this power value, you can determine how much work the engine can perform with a quarter-full gas tank.This conversion ensures accuracy in determining how much the engine can lift, factoring in all forms of energy expenditure including friction.

Potential Energy

Potential energy in the context of lifting is the energy stored due to the object's position relative to Earth. It increases as the bucket is lifted because it's moving against the gravitational force.
Potential energy is calculated as:\[ U = mgh \]where

• \( U \) is potential energy in joules
• \( m \) is mass in kilograms
• \( g \) is gravity (\( 9.8 \, \text{m/s}^2 \))
• \( h \) is height in meters

The engine needs to convert its mechanical energy into potential energy to lift the load.
This informs us that the more potential energy needed, the more work must be done.
Understanding this relationship helps assess whether the engine has sufficient power to achieve the desired height.