🔑:## Step 1: Determine the total mass of the systemTo find the acceleration of the mass, we first need to determine the total mass of the system. This includes the hanging mass (m) and the mass of the wire. Since there are n lines, the total mass of the wire is n * m1 * (2 * dh), considering the wire wraps around the pulleys and has a height difference of dh.## Step 2: Calculate the moment of inertia of the pulley sheavesThe moment of inertia of each pulley sheave is given as Is. Since there are n lines, we need to consider the moment of inertia of all sheaves. However, the problem doesn't specify if the sheaves are connected in a way that their inertias add up directly or if they are independent. For simplicity, we'll consider the total moment of inertia of the sheaves as n * Is.## Step 3: Calculate the moment of inertia of the winch drumThe moment of inertia of the winch drum is given as Jd. This value is directly used in calculations.## Step 4: Apply the gear ratio from the motor to the drumThe gear ratio ng affects how the motor torque Tm is transmitted to the drum. The effective torque at the drum is Tm * ng.## Step 5: Calculate the torque applied to the systemThe torque applied to the system by the motor, considering the gear ratio, is Tm * ng. This torque causes the drum to rotate, which in turn causes the mass to accelerate.## Step 6: Consider the rotational kinematics and dynamicsThe rotational motion of the drum and sheaves is related to the linear motion of the mass. The radius of the sheaves (r) and the drum radius (rd) are crucial in this relationship. The torque applied to the system causes a rotational acceleration, which translates to a linear acceleration of the mass.## Step 7: Derive the equation of motionThe equation of motion for the system can be derived by considering the forces and torques acting on it. The weight of the mass (m * g) acts downward, and the tension in the wire acts upward. The torque from the motor causes the drum to rotate, which increases the tension in the wire. The acceleration of the mass is related to the net force acting on it.## Step 8: Combine rotational and linear dynamicsThe rotational dynamics of the drum and sheaves are connected to the linear dynamics of the mass through the radius of the drum and sheaves. The torque applied to the drum results in a rotational acceleration, which is related to the linear acceleration of the mass by the formula a = α * r, where a is the linear acceleration, α is the angular acceleration, and r is the radius.## Step 9: Formulate the expression for accelerationConsidering the forces, torques, and the relationship between rotational and linear motion, we can formulate an expression for the acceleration of the mass. The net torque acting on the system is Tm * ng, and this torque must overcome the inertia of the drum (Jd), the inertia of the sheaves (n * Is), and the inertia of the mass and wire system. The expression for acceleration (a) will involve the motor torque (Tm), gear ratio (ng), drum radius (rd), moment of inertia of the drum (Jd), moment of inertia of the sheaves (Is), mass of the system (including the wire and the hanging mass), and the gravitational acceleration (g).## Step 10: Write the final expression for accelerationGiven the complexity of the system and the need to consider both rotational and linear dynamics, the expression for acceleration (a) of the mass, when a motor torque (Tm) is applied, can be derived by equating the torque applied to the system with the torque required to accelerate the system. This involves considering the rotational inertia of the drum and sheaves, the linear inertia of the mass and wire, and the gear ratio. The expression would be a = (Tm * ng) / (Jd + n * Is + (m + n * m1 * 2 * dh) * rd^2), considering the rotational to linear acceleration conversion and the effective inertia of the system.The final answer is: boxed{frac{Tm * ng}{Jd + n * Is + (m + n * m1 * 2 * dh) * rd^2}}

🔑:## Step 1: Identify the forces acting on each block and the pulleyFor block m_1, the forces are the tension T_1 in the string and the normal force N from the surface. For block m_2, the forces are the tension T_2 in the string and gravity (m_2 * g). The pulley experiences torque due to the difference in tensions (T_1 - T_2) and has rotational inertia.## Step 2: Apply Newton's second law to block m_1The net force acting on m_1 is T_1, so m_1 * a = T_1, where a is the acceleration of m_1.## Step 3: Apply Newton's second law to block m_2The net force acting on m_2 is m_2 * g - T_2, so m_2 * a = m_2 * g - T_2.## Step 4: Consider the rotational motion of the pulleyThe torque on the pulley is (T_1 - T_2) * R, and the rotational inertia of the pulley is (1/2) * m_p * R^2. The angular acceleration of the pulley is a/R, where a is the linear acceleration of the blocks.## Step 5: Apply the rotational analog of Newton's second law to the pulleyThe torque equals the rotational inertia times the angular acceleration: (T_1 - T_2) * R = (1/2) * m_p * R^2 * (a/R).## Step 6: Simplify the equation from Step 5(T_1 - T_2) = (1/2) * m_p * a.## Step 7: Solve the system of equations from Steps 2, 3, and 6From Step 2: T_1 = m_1 * a.From Step 3: T_2 = m_2 * g - m_2 * a.From Step 6: T_1 - T_2 = (1/2) * m_p * a.Substituting T_1 and T_2 into the equation from Step 6 gives m_1 * a - (m_2 * g - m_2 * a) = (1/2) * m_p * a.This simplifies to m_1 * a + m_2 * a - m_2 * g = (1/2) * m_p * a.Rearranging gives (m_1 + m_2 - (1/2) * m_p) * a = m_2 * g.So, a = m_2 * g / (m_1 + m_2 - (1/2) * m_p).## Step 8: Calculate the tensions T_1 and T_2 using the acceleration found in Step 7T_1 = m_1 * a = m_1 * m_2 * g / (m_1 + m_2 - (1/2) * m_p).T_2 = m_2 * g - m_2 * a = m_2 * g - m_2 * m_2 * g / (m_1 + m_2 - (1/2) * m_p).The final answer is: boxed{a = frac{m_2 g}{m_1 + m_2 - frac{1}{2}m_p}}

🔑:## Step 1: Calculate the number of moles of the fuel mixtureFirst, we need to calculate the total number of moles in the 55-g sample. To do this, we need the molar masses of propane (C3H8) and butane (C4H10). The molar mass of propane is 44.0955 g/mol, and the molar mass of butane is 58.1223 g/mol. Given that the mole fraction of propane is 0.51, the mole fraction of butane is 1 - 0.51 = 0.49.## Step 2: Determine the average molar mass of the mixtureTo find the average molar mass of the mixture, we use the mole fractions and the molar masses of the components. The formula for the average molar mass (M_avg) is: M_avg = ( mole fraction of propane * molar mass of propane ) + ( mole fraction of butane * molar mass of butane ). Substituting the given values: M_avg = (0.51 * 44.0955) + (0.49 * 58.1223).## Step 3: Calculate the average molar massPerforming the calculation: M_avg = (0.51 * 44.0955) + (0.49 * 58.1223) = 22.498505 + 28.5098937 = 50.9983987 g/mol.## Step 4: Calculate the total number of moles in the sampleThe total number of moles (n) in the sample can be found using the formula: n = total mass of the sample / average molar mass of the mixture. Substituting the given values: n = 55 g / 50.9983987 g/mol.## Step 5: Perform the calculation for the total number of molesCalculating the total number of moles: n = 55 g / 50.9983987 g/mol = 1.078 mol.## Step 6: Calculate the mass of propane in the sampleThe mass of propane can be found by multiplying the total number of moles by the mole fraction of propane and then by the molar mass of propane: mass of propane = n * mole fraction of propane * molar mass of propane.## Step 7: Perform the calculation for the mass of propaneSubstituting the values: mass of propane = 1.078 mol * 0.51 * 44.0955 g/mol.## Step 8: Calculate the mass of propanePerforming the calculation: mass of propane = 1.078 mol * 0.51 * 44.0955 g/mol = 23.973 g.## Step 9: Calculate the mass of butane in the sampleThe mass of butane can be found by subtracting the mass of propane from the total mass of the sample, or alternatively, by multiplying the total number of moles by the mole fraction of butane and then by the molar mass of butane: mass of butane = n * mole fraction of butane * molar mass of butane.## Step 10: Perform the calculation for the mass of butane using the alternative methodSubstituting the values: mass of butane = 1.078 mol * 0.49 * 58.1223 g/mol.## Step 11: Calculate the mass of butanePerforming the calculation: mass of butane = 1.078 mol * 0.49 * 58.1223 g/mol = 31.027 g.The final answer is: boxed{23.973}

🔑:To apply periodic boundary conditions to a finite-sized graphene model and simulate an infinite graphene plane, you can follow these steps:1. Define the unit cell: Identify the smallest unit cell of the graphene lattice, which consists of two carbon atoms (A and B) in a hexagonal arrangement.2. Choose the supercell size: Select a supercell size that is a multiple of the unit cell, typically a rectangular or square shape with a large number of unit cells. This will help to minimize the effects of the finite size of the model.3. Apply periodic boundary conditions: Impose periodic boundary conditions on the supercell by identifying the atoms on the edges of the supercell and "connecting" them to the corresponding atoms on the opposite edges. This creates a seamless, infinite lattice.4. Construct the tight-binding Hamiltonian: Write down the tight-binding Hamiltonian for the supercell, including the hopping terms between nearest-neighbor atoms. The Hamiltonian will have a block-diagonal structure, with each block corresponding to a specific momentum vector (k-point) in the Brillouin zone.5. Diagonalize the Hamiltonian: Diagonalize the tight-binding Hamiltonian for each k-point to obtain the energy eigenvalues and eigenvectors. This will give you the band structure of the infinite graphene plane.The implications of this approach on the tight-binding Hamiltonian and the resulting band structure are:* Folded band structure: The band structure of the finite-sized model will be "folded" into the Brillouin zone of the supercell, resulting in a more complex band structure with multiple bands.* Zone-folding: The folding of the band structure leads to the appearance of additional bands, which are replicas of the original bands shifted by reciprocal lattice vectors.* K-point sampling: The choice of k-point sampling will affect the accuracy of the band structure. A finer k-point mesh will provide a more accurate representation of the band structure, but will also increase the computational cost.* Convergence: The band structure will converge to the infinite graphene plane limit as the supercell size increases. However, the rate of convergence may depend on the specific properties being studied, such as the band gap or the density of states.By applying periodic boundary conditions to a finite-sized graphene model, you can simulate an infinite graphene plane and obtain a more accurate representation of its electronic properties, including the band structure and density of states. This approach is widely used in computational materials science and condensed matter physics to study the properties of two-dimensional materials like graphene.