🔑:## Step 1: Determine the minimum velocity required for the cube to complete the loop without friction.To complete the loop, the cube must have enough velocity at the top to ensure that the normal force (N) exerted by the track on the cube is greater than or equal to zero. At the top of the loop, the force of gravity (mg) acts downward, and the normal force acts upward. For the cube to stay on the track, the centrifugal force (mv^2/r) must balance mg, thus mv^2/r = mg. Solving for v gives v = sqrt(rg).## Step 2: Analyze the forces involved in the first half of the loop.In the first half of the loop, the cube experiences a centrifugal force (mv^2/r) outward, a normal force (N) toward the center of the loop, and gravity (mg) downward. The normal force at any point is given by N = (mv^2/r) - mg*cos(θ), where θ is the angle from the top of the loop. However, since we're considering the cube reaching the top with velocity v, we focus on the conditions at the top.## Step 3: Consider the effect of friction on the second half of the loop.When friction is present, it acts opposite to the direction of motion. The force of friction (f) can be given by f = μN, where μ is the coefficient of friction and N is the normal force. At the top of the loop, N = (mv^2/r) - mg. For the cube to continue around the loop with friction, the velocity must be sufficient to overcome the energy loss due to friction.## Step 4: Determine the condition for the cube to remain on the loop with friction.The cube will remain on the loop if the centrifugal force at the top is greater than or equal to the force of gravity, considering the effect of friction. The energy at the top of the loop (E_top) must be enough to overcome the energy loss due to friction (E_friction) over the second half of the loop. E_top = (1/2)mv^2 + mgh, where h = 2r (the height from the bottom of the loop to the top), and E_friction = μmg*cos(θ) * path_length, where path_length for the second half of the loop is πr.## Step 5: Calculate the minimum velocity required with friction.To simplify, we consider the condition at the top of the loop where the cube just stays on the track with friction. The normal force (N) must balance the weight (mg) and provide the centrifugal force, thus N = mg + mv^2/r. With friction, the minimum velocity condition becomes more complex due to the energy considerations over the entire path.## Step 6: Conclude the conditions for the cube to continue around the loop.Without friction, the cube will continue around the loop if its velocity at the top satisfies v >= sqrt(rg). With friction, the condition involves ensuring that the cube has enough initial energy to overcome frictional losses over the second half of the loop, which depends on the coefficient of friction, the mass of the cube, the radius of the loop, and the initial velocity.The final answer is: boxed{v = sqrt{rg}}

🔑:## Step 1: Understand Euler's Equations for a Symmetric TopEuler's equations for a symmetric top are given by:[I_1 dot{omega}_1 + (I_3 - I_2) omega_2 omega_3 = 0][I_2 dot{omega}_2 + (I_1 - I_3) omega_3 omega_1 = 0][I_3 dot{omega}_3 + (I_2 - I_1) omega_1 omega_2 = 0]where (I_1), (I_2), and (I_3) are the moments of inertia about the principal axes, and (omega_1), (omega_2), and (omega_3) are the angular velocities about these axes.## Step 2: Consider Small Variations in Angular VelocitiesFor a symmetric top like a coin, we have (I_1 = I_2 neq I_3). Assuming the coin is spinning about its symmetric axis (axis 3), we have (omega_3) as the initial angular velocity, and (omega_1 = omega_2 = 0) initially. We consider small variations in (omega_1) and (omega_2), denoted as (deltaomega_1) and (deltaomega_2), to analyze the stability of the spin.## Step 3: Linearize Euler's EquationsSubstituting (I_1 = I_2) and considering (omega_1 = deltaomega_1), (omega_2 = deltaomega_2), and (omega_3 = Omega + deltaomega_3), where (Omega) is the constant angular velocity about the symmetric axis, into Euler's equations, we get:[I_1 deltadot{omega}_1 + (I_3 - I_1) deltaomega_2 Omega = 0][I_1 deltadot{omega}_2 + (I_1 - I_3) deltaomega_1 Omega = 0][I_3 deltadot{omega}_3 = 0]Since (deltaomega_3) is not relevant to the stability about the spin axis, we focus on the first two equations.## Step 4: Solve the Linearized EquationsTo solve these equations, we look for solutions of the form (deltaomega_1 = A e^{ilambda t}) and (deltaomega_2 = B e^{ilambda t}), where (A), (B), and (lambda) are constants. Substituting these into the linearized equations gives:[ilambda I_1 A + (I_3 - I_1) Omega B = 0][ilambda I_1 B + (I_1 - I_3) Omega A = 0]From these, we can derive the characteristic equation:[lambda^2 = frac{(I_3 - I_1)(I_1 - I_3)}{I_1^2} Omega^2]## Step 5: Determine Stability ConditionsFor stability, we require the solutions to be bounded over time, meaning (lambda^2 leq 0). However, since (lambda^2) is proportional to ((I_3 - I_1)^2 Omega^2 / I_1^2), which is always non-negative, the condition for stability in terms of the moments of inertia and initial angular velocity cannot be directly derived from this simplified analysis. Instead, we must consider the nature of the motion and the effects of damping.## Step 6: Consider Aerodynamic Drag and Moment of Inertia EffectsAerodynamic drag acts as a damping force, tending to reduce the angular velocity over time. The moment of inertia (I_3) about the spin axis affects the stability; a larger (I_3) compared to (I_1) and (I_2) generally leads to more stable spinning due to the conservation of angular momentum. However, the mathematical derivation of stability conditions must incorporate these factors in a more nuanced analysis, considering the specific dynamics of the coin's motion and the effects of air resistance.## Step 7: Conclusion on StabilityGiven the simplifications and the nature of Euler's equations for a symmetric top, the stability of the coin's spin is influenced by its moments of inertia and the initial conditions of the spin. Aerodynamic drag plays a crucial role in damping out oscillations and stabilizing the spin over time. However, the precise mathematical conditions for stability, incorporating these factors, require a more detailed analysis that accounts for the specific dynamics of the spinning coin and the effects of air resistance.The final answer is: boxed{I_3 > I_1}

🔑:To solve this problem, we'll apply Newton's second law of motion to each block and consider the forces acting on them.## Step 1: Determine the forces acting on each blockFor block m1 on the inclined plane, the forces are the tension (T) in the cord acting up the incline, the weight (m1g) of the block acting downward, and the normal force (N) from the plane acting perpendicular to the incline. For block m2, the forces are its weight (m2g) acting downward and the tension (T) in the cord acting upward.## Step 2: Resolve the forces into components for block m1The weight of block m1 can be resolved into two components: one perpendicular to the incline (m1g*cos(30°)) and one parallel to the incline (m1g*sin(30°)). The tension (T) acts up the incline, opposing the component of m1's weight that is parallel to the incline.## Step 3: Apply Newton's second law to block m1The net force acting on block m1 along the incline is T - m1g*sin(30°). According to Newton's second law, this net force equals the mass of block m1 times its acceleration (a): T - m1g*sin(30°) = m1a.## Step 4: Apply Newton's second law to block m2For block m2, the net force is m2g - T, and this equals the mass of block m2 times its acceleration (a): m2g - T = m2a.## Step 5: Equate the accelerations and solve for T and aSince the accelerations of both blocks are the same (a), we can set the equations from steps 3 and 4 equal to each other to find T and then solve for a.Given:- m1 = 3.6 kg- m2 = 2.1 kg- g = 9.81 m/s^2 (acceleration due to gravity)- Angle = 30 degreesFirst, calculate sin(30°) and cos(30°):- sin(30°) = 0.5- cos(30°) = √3/2## Step 6: Solve the equationsFrom step 3: T - m1g*sin(30°) = m1aFrom step 4: m2g - T = m2aSince a is the same for both blocks, we can set the two equations equal to each other:T - m1g*sin(30°) = m2g - TRearrange to solve for T:2T = m2g + m1g*sin(30°)T = (m2g + m1g*sin(30°)) / 2Substitute known values:T = (2.1*9.81 + 3.6*9.81*0.5) / 2T = (20.61 + 17.64) / 2T = 38.25 / 2T = 19.125 NNow, substitute T back into one of the original equations to solve for a:Using T - m1g*sin(30°) = m1a19.125 - 3.6*9.81*0.5 = 3.6a19.125 - 17.646 = 3.6a1.479 = 3.6aa = 1.479 / 3.6a = 0.4111 m/s^2The final answer is: boxed{0.4111}

🔑:Implementing a new public transportation system is a complex and multifaceted project that requires careful planning and consideration of various factors. Conducting a thorough SWOT (Strengths, Weaknesses, Opportunities, and Threats) analysis is crucial in informing the strategic direction of this project. A SWOT analysis helps identify the internal and external factors that can impact the project's success, allowing the local government to make informed decisions and develop effective strategies to address potential challenges.Importance of SWOT Analysis:1. Identifying Strengths and Weaknesses: A SWOT analysis helps identify the local government's internal strengths and weaknesses, such as existing infrastructure, funding, and expertise. This information can be used to leverage strengths and address weaknesses, ensuring the project's success.2. Recognizing Opportunities and Threats: The analysis also identifies external opportunities and threats, such as changing demographics, technological advancements, and potential opposition from stakeholders. This information can be used to capitalize on opportunities and mitigate threats.3. Informing Strategic Direction: The SWOT analysis provides a comprehensive understanding of the project's internal and external environment, enabling the local government to develop a strategic plan that addresses potential challenges and leverages opportunities.Impact on Stakeholders:1. Residents and Commuters: A new public transportation system can significantly impact residents and commuters, affecting their daily lives, travel times, and overall quality of life. A SWOT analysis can help identify potential benefits and drawbacks, ensuring that the project meets the needs of these stakeholders.2. Businesses and Local Economy: The project can also impact local businesses, affecting their accessibility, customer base, and overall economic viability. A SWOT analysis can help identify opportunities to support local businesses and stimulate economic growth.3. Environmental and Social Impacts: The project's environmental and social impacts, such as air pollution, noise, and displacement of communities, must also be considered. A SWOT analysis can help identify potential risks and opportunities to mitigate them.Role of Political Responsiveness:1. Public Engagement: Political responsiveness requires engaging with the public and stakeholders to understand their needs, concerns, and expectations. A SWOT analysis can inform this engagement process, ensuring that the project is responsive to the community's needs.2. Balancing Competing Interests: The local government must balance competing interests and priorities, such as economic development, environmental sustainability, and social equity. A SWOT analysis can help identify potential trade-offs and inform decision-making.3. Building Support and Legitimacy: A SWOT analysis can help build support and legitimacy for the project by demonstrating a thorough understanding of the project's potential impacts and a commitment to addressing stakeholder concerns.Ethical Considerations:1. Equity and Accessibility: The project must ensure equity and accessibility for all members of the community, including low-income residents, people with disabilities, and marginalized groups. A SWOT analysis can help identify potential barriers and opportunities to address them.2. Environmental Sustainability: The project's environmental impacts, such as greenhouse gas emissions and resource consumption, must be considered and mitigated. A SWOT analysis can help identify opportunities to incorporate sustainable design and operations.3. Transparency and Accountability: The local government must ensure transparency and accountability throughout the project's development and implementation. A SWOT analysis can help identify potential risks and opportunities to ensure transparency and accountability.In conclusion, conducting a thorough SWOT analysis is essential for informing the strategic direction of a new public transportation system. By considering the potential impact on stakeholders, the role of political responsiveness, and the ethical considerations involved, the local government can develop a project that meets the needs of the community, promotes sustainability, and ensures equity and accessibility. A SWOT analysis provides a comprehensive understanding of the project's internal and external environment, enabling the local government to make informed decisions and develop effective strategies to address potential challenges and capitalize on opportunities.