🔑:To solve this problem, we'll follow a step-by-step approach considering the solubility of each salt in water at 100°C, and then compare the composition of the solution when both salts are dissolved together versus separately.## Step 1: Determine the Solubility of NaCO3 and KCO3 in Water at 100°CAt 100°C, the solubility of NaCO3 (sodium carbonate) is approximately 30 g/100 mL, and the solubility of KCO3 (potassium carbonate) is approximately 137 g/100 mL in water. However, these values can vary slightly based on the specific conditions and sources. For simplicity, we'll use these approximate values.## Step 2: Calculate the Maximum Amount of Each Salt That Can Dissolve in 1 Liter of WaterFor NaCO3: 30 g/100 mL * 1000 mL/L = 300 g/LFor KCO3: 137 g/100 mL * 1000 mL/L = 1370 g/L## Step 3: Calculate the Total Amount of Each Salt AvailableGiven: 1.5 kg (1500 g) of NaCO3 and 1.5 kg (1500 g) of KCO3.## Step 4: Determine the Composition of the Saturated Solution When Both Salts Are Dissolved TogetherSince both salts are highly soluble and we have a mixture, we need to consider if there's any interaction or limit to how much of each can dissolve in the presence of the other. In general, for salts like NaCO3 and KCO3, their solubility in water is not significantly affected by the presence of each other in terms of forming precipitates or common ion effects that would limit solubility, given their different cations and anions. However, the solubility product constant (Ksp) and the common ion effect could play a role in more complex systems. For simplicity, assuming ideal behavior and no significant interaction affecting solubility, we can consider the maximum solubility of each salt as a guideline.## Step 5: Compare the Composition of the Solution When Each Salt Is Dissolved SeparatelyWhen dissolved separately, each salt would reach its maximum solubility in 1 liter of water as calculated in Step 2. However, when mixed, the total amount of salt (3000 g) far exceeds the solubility of either salt alone, indicating that the solution will be saturated with respect to at least one of the salts.## Step 6: Determine the Limiting Factor for the Composition of the Saturated SolutionGiven the high solubility of KCO3 compared to NaCO3, and considering the amounts available (1500 g each), the limiting factor will be the solubility of NaCO3, as it is less soluble than KCO3. The solution will be saturated with respect to NaCO3 first.## Step 7: Calculate the Composition of the Saturated SolutionSince NaCO3 limits the solubility, it will fully dissolve up to its solubility limit (300 g/L), and the remaining amount will be undissolved. KCO3, being more soluble, will continue to dissolve until it reaches its solubility limit or until the solution is saturated with respect to both salts.The final answer is: boxed{300}

🔑:## Step 1: Calculate the moment of inertia using the initial and final kinetic energies and angular velocities.First, we use the formula for kinetic energy, (KE = frac{1}{2}Iomega^2), where (I) is the moment of inertia and (omega) is the angular velocity. Given (KE_{initial} = 11.52) J and (omega_{initial} = 3) rad/sec, and (KE_{final} = 184.32) J and (omega_{final} = 48) rad/sec, we can set up the equations: (11.52 = frac{1}{2}I(3)^2) and (184.32 = frac{1}{2}I(48)^2). Solving for (I) in both equations will help us find a relationship or confirm the moment of inertia.## Step 2: Solve for the moment of inertia using the initial conditions.From (11.52 = frac{1}{2}I(3)^2), we get (11.52 = frac{1}{2}I(9)), which simplifies to (11.52 = 4.5I). Solving for (I), we find (I = frac{11.52}{4.5}).## Step 3: Calculate the moment of inertia.(I = frac{11.52}{4.5} = 2.56) kg*m^2.## Step 4: Verify the moment of inertia using the final conditions.Using (184.32 = frac{1}{2}I(48)^2), we get (184.32 = frac{1}{2}I(2304)), which simplifies to (184.32 = 1152I). Solving for (I), we find (I = frac{184.32}{1152}).## Step 5: Calculate the moment of inertia from the final conditions.(I = frac{184.32}{1152} = 0.16) kg*m^2. However, this step seems to introduce an inconsistency with the calculation from the initial conditions. The correct approach should involve using the conservation of angular momentum to find the relationship between the initial and final states, not directly calculating (I) from final conditions without considering the change in radius.## Step 6: Apply the principle of conservation of angular momentum.The angular momentum (L = Iomega) is conserved. Given (L = 7.68), we can relate the initial and final states: (I_{initial}omega_{initial} = I_{final}omega_{final}). Since (I = mr^2), and knowing the radii, we can relate the initial and final moments of inertia and angular velocities.## Step 7: Correctly relate the initial and final moments of inertia and angular velocities.Given (r_{initial} = 0.8) m and (r_{final} = 0.2) m, and knowing (omega_{initial} = 3) rad/sec and (omega_{final} = 48) rad/sec, we use (L = Iomega = mr^2omega) to find the relationship between the initial and final states, considering (L) is constant.## Step 8: Use the conservation of angular momentum to find the mass of the dumbbell.Since (L = mr_{initial}^2omega_{initial} = mr_{final}^2omega_{final}), we can solve for (m). Given (L = 7.68), (r_{initial} = 0.8) m, (r_{final} = 0.2) m, (omega_{initial} = 3) rad/sec, and (omega_{final} = 48) rad/sec, we substitute these values into the equation.## Step 9: Calculate the mass of the dumbbell.Rearranging (mr_{initial}^2omega_{initial} = mr_{final}^2omega_{final}) to solve for (m) is not necessary since (L = 7.68 = mr_{initial}^2omega_{initial}). We can directly find (m) using the given (L) and initial conditions: (7.68 = m(0.8)^2(3)).## Step 10: Solve for the mass of the dumbbell.(7.68 = m(0.64)(3)) simplifies to (7.68 = 1.92m). Solving for (m), we get (m = frac{7.68}{1.92}).## Step 11: Calculate the mass.(m = frac{7.68}{1.92} = 4) kg.## Step 12: Find the force required to pull one of the dumbbells in at a constant speed.The force (F) required is related to the change in kinetic energy and the distance over which the force is applied. Since the dumbbell is pulled from 0.8 m to 0.2 m, the distance (d = 0.8 - 0.2 = 0.6) m. The change in kinetic energy (Delta KE = KE_{final} - KE_{initial}).## Step 13: Calculate the change in kinetic energy.(Delta KE = 184.32 - 11.52 = 172.8) J.## Step 14: Calculate the force required.The force (F = frac{Delta KE}{d}), since the work done (W = Fd = Delta KE).## Step 15: Calculate the force.(F = frac{172.8}{0.6}).The final answer is: boxed{288}

🔑:## Step 1: Understand the scenarioA box is moving with a constant velocity u on a smooth horizontal surface, and sand is being removed from it. The removed sand has the same horizontal velocity u as the box.## Step 2: Identify the forces acting on the boxSince the box is moving on a smooth horizontal surface, there is no frictional force acting on it. The only forces to consider are the force due to the removal of sand and any external forces acting on the box.## Step 3: Apply the formula F = dp/dt to the boxThe formula F = dp/dt relates the net force acting on an object to its rate of change of momentum. For the box, the momentum is changing because sand is being removed, which carries away some of the box's momentum.## Step 4: Consider the momentum of the removed sandThe sand being removed from the box carries away momentum. Since the sand has the same horizontal velocity u as the box, the momentum carried away by the sand is in the same direction as the box's velocity.## Step 5: Analyze the apparent paradoxThe apparent paradox arises because the formula F = dp/dt suggests that there should be a force acting on the box in the direction of its velocity, due to the change in momentum. However, since the box is moving with constant velocity, there is no net external force acting on it.## Step 6: Resolve the paradox using the concept of a 'system'The key to resolving the paradox is to consider the box and the sand as separate parts of the system. When sand is removed from the box, it carries away momentum, but this does not directly affect the box's velocity because the sand and the box are considered as separate entities within the system.## Step 7: Reconsider the application of F = dp/dtWhen applying F = dp/dt to the box as a system, we must consider the change in momentum of the entire system, including both the box and the removed sand. Since the removed sand carries away momentum, the total momentum of the system (box + sand) remains constant, and thus the net force acting on the system is zero.## Step 8: ConclusionThe apparent paradox is resolved by recognizing that the formula F = dp/dt applies to the entire system, including both the box and the removed sand. By considering the system as a whole, we can see that the change in momentum of the box due to the removal of sand is balanced by the momentum carried away by the sand, resulting in no net force acting on the system.The final answer is: boxed{0}

🔑:A 1994 Cadillac Deville with a leaking engine block! That's a serious issue. Let's dive into the potential causes, diagnosis, and repair steps to help you get your car back on the road.Potential Causes:1. Cracked Engine Block: Overheating, excessive pressure, or manufacturing defects can cause the engine block to crack, leading to water leakage.2. Head Gasket Failure: A blown head gasket can allow coolant to escape from the engine block, causing water to pour out.3. Intake Manifold Gasket Leak: A leaking intake manifold gasket can also allow coolant to leak from the engine block.4. Water Pump Failure: A faulty water pump can cause coolant to leak from the engine block, especially if the pump's shaft seal is damaged.5. Radiator Hose or Connection Issues: A loose, damaged, or corroded radiator hose or connection can cause coolant to leak from the engine block.6. Cylinder Head or Engine Block Damage: Damage to the cylinder head or engine block, such as a cracked or warped surface, can cause water to leak from the engine block.7. Coolant System Clog: A clogged coolant system, such as a blocked radiator or heater core, can cause pressure to build up and lead to water leakage from the engine block.Diagnosis Steps:1. Visual Inspection: Look for signs of water leakage, such as puddles, drips, or stains, around the engine block, radiator, and hoses.2. Check Coolant Level: Monitor the coolant level to see if it's dropping over time.3. Pressure Test: Perform a pressure test on the cooling system to identify any leaks or blockages.4. Compression Test: Conduct a compression test to check for any internal engine issues, such as a blown head gasket or cracked engine block.5. Leak-Down Test: Perform a leak-down test to identify any leaks in the engine block or cylinder head.6. Inspect Hoses and Connections: Check all radiator hoses, connections, and clamps for signs of damage, corrosion, or wear.7. Scan for Trouble Codes: Use a scan tool to check for any trouble codes related to the cooling system or engine performance.Repair Steps:1. Replace Cracked Engine Block: If the engine block is cracked, it may need to be replaced, which can be a costly and time-consuming process.2. Replace Head Gasket: If the head gasket is blown, replace it with a new one, and ensure the cylinder head is properly resurfaced or replaced if necessary.3. Replace Intake Manifold Gasket: If the intake manifold gasket is leaking, replace it with a new one, and ensure the intake manifold is properly sealed.4. Replace Water Pump: If the water pump is faulty, replace it with a new one, and ensure the coolant system is properly flushed and filled.5. Replace Radiator Hoses or Connections: If a radiator hose or connection is damaged, replace it with a new one, and ensure all connections are secure and tightened properly.6. Repair or Replace Cylinder Head: If the cylinder head is damaged, repair or replace it as needed, and ensure the engine block is properly resurfaced or replaced if necessary.7. Flush and Fill Coolant System: Once the repair is complete, flush and fill the coolant system with a 50/50 mix of antifreeze and water, and ensure the system is properly bled to remove any air pockets.Additional Tips:* Always consult a repair manual or a professional mechanic if you're unsure about any aspect of the diagnosis or repair process.* Use a pressure test kit and a leak-down test kit to help identify the source of the leak.* Consider using a coolant system sealant to help stop small leaks, but be aware that this may not be a permanent fix.* Keep in mind that a leaking engine block can cause significant damage to the engine and other components, so it's essential to address the issue promptly to avoid further damage.Remember, a proper diagnosis and repair require patience, attention to detail, and a thorough understanding of the engine and cooling system. If you're not comfortable with the repair process, it's always best to consult a professional mechanic for assistance.