91Ó°ÊÓ

A boy of mass \(m\) is standing initially at rest relative to the moving walkway inclined at the angle \(\theta\) and moving with a constant speed \(u .\) He decides to accelerate his progress and starts to walk from point \(A\) with a steadily increasing speed and reaches point \(B\) with a speed \(v_{r}\) relative to the walkway. During his acceleration he generates a constant average force \(F\) tangent to the walkway between his shoes and the walkway surface. Write the work-energy equations for the motion between \(A\) and \(B\) for his absolute motion and his relative motion and explain the meaning of the term \(m u v_{r}\) If the boy weighs 150 lb and if \(u=2 \mathrm{ft} / \mathrm{sec}, s=\) \(30 \mathrm{ft},\) and \(\theta=10^{\circ},\) calculate the power \(P_{\mathrm{rel}}\) developed by the boy as he reaches the speed of \(2.5 \mathrm{ft} / \mathrm{sec}\) relative to the walkway.

Step by step solution

01

Understanding Absolute Motion

Absolute motion considers the velocity of the boy relative to a stationary observer. Therefore, we add the boy's velocity relative to the walkway, \(v_r\), to the speed of the walkway, \(u\). Thus, the boy's absolute velocity \(v_{abs}\) at point B is given by \(v_{abs} = u + v_r\).

02

Writing Work-Energy Equation for Absolute Motion

Apply the work-energy principle for absolute motion: \(T_A + W_{AB} = T_B\), where \(T\) represents the kinetic energy. Initially, the kinetic energy \(T_A\) at point A is 0 because he starts from rest (\(v_{abs} = 0\)). The work done \(W_{AB}\) is \(Fs\) where \(F\) is the force and \(s\) is the distance. At point B, the kinetic energy \(T_B\) is \(\frac{1}{2}m(v_{abs})^2\). Substitute known values into the equation: \(Fs = \frac{1}{2}m(u + v_r)^2\).

03

Writing Work-Energy Equation for Relative Motion

For relative motion, consider only the velocity relative to the walkway. The kinetic energy at point A is still 0. Therefore, the work-energy equation is: \(Fs = \frac{1}{2}m(v_r)^2\). This shows that the work done is used entirely to give him speed \(v_r\) relative to the walkway.

04

Understanding the term \(muv_r\)

The term \(muv_r\) represents the cross-term in the expansion of \((u + v_r)^2\). Physically, it represents the additional kinetic energy component due to the boy moving relative to a moving frame (the walkway). It accounts for the interaction between the boy’s velocity relative to the walkway and the walkway’s constant speed.

05

Calculating Power Developed Relative to the Walkway

Power is the rate of work done, or \(P = \frac{W}{t}\). The work done relative to the walkway is \(W = Fs = 0.5m(v_r)^2\). First, convert 150 lb to mass in slug using \(1 ext{ lb} = 0.03108 ext{ slug}\), giving approximately \(m = 150 imes 0.03108 = 4.662 ext{ slugs}\). Use the relative work-energy equation \(Fs = 0.5(4.662)(2.5)^2\). Solve for \(F\) given \(s = 30\). Substitute \(F\) back into \(Fv_r\) to find the power \(P_{rel}\), since \(P_{rel} = Fv_r\). Substitute known values for \(v_r\) and calculate \(P_{rel}\).

06

Final Calculation

Using the earlier values and \(F\) found: Compute \(F = \frac{0.5(4.662)(2.5)^2}{30} = \frac{7.2875}{30}\) to get approximately \(F = 0.2429\text{ lbf}\). So, \(P_{rel} = Fv_r = (0.2429)(2.5)\) is approximately \(0.607 ext{ ft-lb/sec}\).

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Key Concepts

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

Relative Motion

Relative motion refers to the movement of an object as observed from another moving reference point. In the case of the boy on the walkway, his movement is observed relative to the walkway itself. When we describe his speed as being "relative," we mean it is how fast he moves considering the walkway’s constant speed.

• If he stands still relative to the earth, he moves at speed \(u\) due to the walkway.
• If he walks forward while the walkway moves, his relative speed \(v_r\) adds on to the walkway's speed \(u\).

When analyzing problems involving relative motion, always distinguish between absolute motion (as seen from the ground) and relative motion (as seen from another moving object). This distinction is crucial in dynamics and can change how forces and energies are perceived.

Kinetic Energy

Kinetic energy is the energy of motion. For any object, including the boy on the walkway, it is calculated as \(T = \frac{1}{2} m v^2\), where \(m\) is the mass and \(v\) is the velocity.

For the boy:

• Absolute kinetic energy considers his comprehensive velocity, \(v_{ ext{abs}} = u + v_{r}\), adding both the walkway's and his relative velocity.
• Relative kinetic energy is simpler, focusing just on his speed compared to the walkway, \(v_r\).

The kinetic energy at any point gives insight into how much work (or energy) has been used to reach that pace from rest. It's essential in systems where speeds and forces vary during motion.

Mechanical Power

Mechanical power is the rate at which work is done. If the boy exerts force \(F\) to move faster, then he performs work over time, producing power. It’s calculated by \(P = F v\), where \(F\) is the force applied, and \(v\) is the speed at which force acts.

For relative motion on the walkway:

• The power developed is determined by the relative velocity \(v_r\).
• This power reflects how quickly energy is expended to keep accelerating.

Understanding mechanical power helps in assessing the efficiency of the motion. When you know \(P\), you can determine if a system is exerting excessive energy or is optimally tuned.

Work Done

Work done refers to the energy transfer that occurs when a force moves an object over a distance. When the boy on the walkway exerts force to increase his speed, he does work, which can be written as \(W = F s\), where \(s\) is the distance.

In the scenario:

• The work done against the walkway helps him achieve a new speed \(v_r\).
• This work translates into changes in kinetic energy, either relative or absolute.

Understanding work done is critical, as it ties the force, distance, and energy change together. It helps clarify how much effort is required to achieve specific speed changes, reflecting energy efficiency in physical systems.