🔑:## Step 1: Understanding the Strong Nuclear Force and Electromagnetic RepulsionThe strong nuclear force is a short-range attractive force that holds protons and neutrons together in the nucleus of an atom, overcoming the electromagnetic repulsion between positively charged protons. This force is responsible for the stability of atomic nuclei.## Step 2: Binding Energy Curve ExplanationThe binding energy curve is a graphical representation of the binding energy per nucleon versus the atomic mass number (A) of an element. It shows that the binding energy per nucleon increases as the atomic mass number increases up to a point (around iron-56 and nickel-56), and then it decreases. This curve indicates the stability of nuclei, with higher binding energy per nucleon corresponding to greater stability.## Step 3: Fusion Above Nickel-56Fusion is the process by which two light atomic nuclei combine to form a heavier nucleus. For nuclei lighter than nickel-56, fusion releases energy because the resulting nucleus has a higher binding energy per nucleon than the original nuclei. However, for nuclei heavier than nickel-56, fusion requires energy because the resulting nucleus has a lower binding energy per nucleon.## Step 4: Interplay Between Strong Nuclear Force and Electromagnetic RepulsionAs nuclei get heavier, the strong nuclear force, which attracts nucleons together, becomes less effective compared to the electromagnetic repulsion, which pushes protons apart. For nuclei heavier than nickel-56, the electromagnetic repulsion dominates, making it more difficult to add more protons to the nucleus without requiring additional energy to overcome this repulsion.## Step 5: Implications for Nuclear StabilityThe need for energy to fuse nuclei heavier than nickel-56 implies that these nuclei are less stable than those around the peak of the binding energy curve (like iron-56 and nickel-56). This instability means that heavier nuclei tend to undergo radioactive decay or other processes to move towards more stable configurations, releasing energy in the process.The final answer is: boxed{Fusion above nickel-56 requires energy because the resulting nucleus has a lower binding energy per nucleon, due to the dominance of electromagnetic repulsion over the strong nuclear force for heavier nuclei.}

🔑:## Step 1: Understand the Kinetic Theory of GasesThe kinetic theory of gases assumes that a gas is composed of a large number of molecules in constant random motion. The molecules are considered to be point particles with no intermolecular forces except during elastic collisions.## Step 2: Define the Variables and Constants- (P) is the pressure of the gas.- (V) is the volume of the gas.- (n) is the number of moles of gas.- (R) is the gas constant.- (T) is the temperature of the gas in Kelvin.- (N) is the total number of molecules.- (N_A) is Avogadro's number, relating the number of molecules to the number of moles.- (m) is the mass of a single molecule.- (v_{rms}) is the root-mean-square velocity of the gas molecules.## Step 3: Derive the Expression for PressureThe pressure exerted by a gas on a wall is due to the collisions of the gas molecules with the wall. The force exerted by a single molecule during a collision is (2mv_{x}), where (v_{x}) is the velocity component perpendicular to the wall. For a large number of molecules, the mean force exerted on a unit area (pressure) can be calculated by considering the number of collisions per unit time and the change in momentum per collision.## Step 4: Calculate the Number of Collisions per Unit TimeThe number of molecules colliding with a unit area of the wall per unit time is given by (frac{1}{2}N/V cdot v_{rms} cdot 1), where (N/V) is the number density of molecules, and the factor (frac{1}{2}) accounts for the fact that only half of the molecules are moving towards the wall at any given time.## Step 5: Calculate the Change in Momentum per CollisionThe change in momentum per collision for a single molecule is (2mv_{x}). However, to relate this to the rms velocity, we use the fact that (v_{rms}^2 = frac{3kT}{m}), where (k) is Boltzmann's constant, and (T) is the temperature. The average value of (v_{x}^2) is (frac{1}{3}v_{rms}^2) due to the equipartition of energy among the three dimensions.## Step 6: Derive the Expression for PressureThe pressure (P) is given by the force per unit area, which is the number of collisions per unit time multiplied by the change in momentum per collision. This yields (P = frac{1}{3} frac{N}{V} mv_{rms}^2). Since (v_{rms}^2 = frac{3kT}{m}), we substitute to get (P = frac{N}{V} kT).## Step 7: Relate the Number of Molecules to MolesThe number of molecules (N) can be expressed in terms of the number of moles (n) and Avogadro's number (N_A) as (N = nN_A). Substituting this into the expression for pressure gives (P = frac{nN_A}{V} kT).## Step 8: Introduce the Gas ConstantThe gas constant (R) is defined as (R = N_Ak), where (k) is Boltzmann's constant. Substituting (R) into the expression for pressure yields (P = frac{nR}{V} T), or (PV = nRT), which is the ideal gas law.The final answer is: boxed{PV = nRT}

🔑:## Step 1: Identify all forces acting on the rockThe rock is subject to two main forces: the force applied upwards by the rope (100 N) and the force of gravity acting downwards. The force of gravity can be calculated using the formula F = mg, where m is the mass of the rock (10.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).## Step 2: Calculate the force of gravity acting on the rockF_gravity = m * g = 10.0 kg * 9.8 m/s^2 = 98 N.## Step 3: Determine the net force acting on the rockThe net force (F_net) is the difference between the applied force (100 N) and the force of gravity (98 N), since these forces are acting in opposite directions. F_net = F_applied - F_gravity = 100 N - 98 N = 2 N.## Step 4: Calculate the acceleration of the rockUsing Newton's second law of motion, F = ma, where F is the net force acting on the rock (2 N), m is the mass of the rock (10.0 kg), and a is the acceleration we are solving for. Rearranging the formula to solve for acceleration gives a = F / m.## Step 5: Calculate the accelerationa = F_net / m = 2 N / 10.0 kg = 0.2 m/s^2.The final answer is: boxed{0.2}

🔑:to 1:The total return is composed of the reinvested coupons and the sale price of the bond at the end of the investment horizon. The future value of the reinvested coupons is calculated as follows:[text{FV}=45timesleft[frac{left(1+frac{0.084}{2}right)^{10}-1}{frac {0.084}{2}}right]=45times 10.824=487.08]The price of the bond at the end of the investment horizon (when it becomes a 2-year bond) can be calculated as the present value of the remaining payments:[text{PV}=45timesleft[frac{1-left(1+frac{0.112}{2}right)^{-4}}{frac {0.112}{2}}right]+frac{1,000}{left(1+frac{0.112}{2}right)^{4}}=45 times 3.387+frac{1,000}{1.243}= 152.66+804.85=957.51]Therefore, the total return is 487.08 + 957.51 = 1,444.59.The holding period return is:[text{HPR}=frac{1,444.59}{1,000}-1=44.459%]The _annualized_ holding period return is:[left(1+text{HPR}right)^{nicefrac{{1}}{{5}}}-1=left(1.44459right)^{0.2 }-1=7.79%]Note that the total return and the holding period return are greater than the yield to maturity at the time of purchase. This occurs because the coupons are reinvested at a rate greater than the yield to maturity of the bond and the bond is sold at a price greater than its par value.