🔑:## Step 1: Understand the given problem and the components involvedWe have a resistive loop placed inside a solenoid. The magnetic flux through the loop due to the solenoid is given by Φ1 = MI1, where M is the mutual inductance and I1 is the current through the solenoid. The magnetic flux through the loop due to itself is Φ2 = LI2, where L is the self-inductance and I2 is the current through the loop. We need to derive the differential equation that governs I2(t) in terms of I1(t), M, L, and R.## Step 2: Apply Faraday's law of induction to the loopFaraday's law of induction states that the induced electromotive force (EMF) in a loop is equal to the negative rate of change of the magnetic flux through the loop. The total magnetic flux through the loop is the sum of Φ1 and Φ2. Therefore, the induced EMF (ε) is given by ε = -d(Φ1 + Φ2)/dt.## Step 3: Express the induced EMF in terms of the given quantitiesSubstituting Φ1 = MI1 and Φ2 = LI2 into the equation for ε, we get ε = -d(MI1 + LI2)/dt = -M(dI1/dt) - L(dI2/dt).## Step 4: Relate the induced EMF to the current through the loop using Ohm's lawThe induced EMF (ε) is also equal to the voltage drop across the resistance of the loop, which by Ohm's law is given by ε = -RI2.## Step 5: Equate the two expressions for ε and solve for dI2/dtEquating the expressions for ε from steps 3 and 4, we have -M(dI1/dt) - L(dI2/dt) = -RI2. Rearranging this equation to solve for dI2/dt gives dI2/dt = (R/L)I2 - (M/L)(dI1/dt).## Step 6: Write the differential equation governing I2(t)The differential equation that governs the current through the loop, I2(t), in terms of I1(t), M, L, and R is therefore dI2/dt + (R/L)I2 = (M/L)(dI1/dt).The final answer is: boxed{frac{dI_2}{dt} + frac{R}{L}I_2 = frac{M}{L}frac{dI_1}{dt}}

🔑:## Step 1: Understanding the Scenario Without Air ResistanceWhen a ball is thrown upwards and air resistance is ignored, the ball's motion can be described using the principles of projectile motion under the sole influence of gravity. The initial velocity imparted to the ball has two components: horizontal and vertical. Since there's no air resistance, the horizontal component of velocity remains constant throughout the flight. The vertical component of velocity, however, is affected by gravity, which accelerates the ball downwards at a rate of 9.81 m/s^2 (on Earth).## Step 2: Position, Velocity, and Speed Without Air Resistance- Position: The ball initially moves upwards, reaches a maximum height where its vertical velocity is momentarily zero, and then descends back to its starting point.- Velocity: The velocity decreases as the ball goes up (due to gravity) and increases as it comes down. At the peak, the vertical velocity is zero.- Speed: Speed, being the magnitude of velocity, decreases as the ball ascends (since the vertical component of velocity decreases), reaches a minimum at the peak (where only the horizontal component of velocity exists if the throw was not perfectly vertical), and then increases as the ball descends.- Acceleration: The acceleration due to gravity is constant and directed downwards, at 9.81 m/s^2. This acceleration affects only the vertical component of the ball's velocity.## Step 3: Understanding the Scenario With Air ResistanceWhen air resistance is taken into account, the ball's motion becomes more complex. Air resistance opposes the motion of the ball, acting in the opposite direction to the ball's velocity. This means it slows down the ball both as it ascends and as it descends, but its effect is more pronounced when the ball is moving faster (i.e., during the descent).## Step 4: Position, Velocity, and Speed With Air Resistance- Position: The ball still follows a parabolic path, but air resistance reduces the maximum height reached and the range of the projectile compared to a resistance-free scenario.- Velocity: Air resistance decreases the ball's velocity throughout its flight, both upwards and downwards, but the effect is more significant during the descent due to the higher speeds.- Speed: The speed of the ball decreases more rapidly as it ascends due to both gravity and air resistance. During descent, while gravity increases the speed, air resistance works to decrease it, resulting in a net increase in speed but at a reduced rate compared to a scenario without air resistance.- Acceleration: The acceleration of the ball is no longer constant due to the variable effect of air resistance. The downwards acceleration due to gravity is still 9.81 m/s^2, but there's an additional acceleration opposite to the direction of motion due to air resistance, which increases with the square of the velocity.## Step 5: Physical Principles Supporting the ExplanationsThe physical principles supporting these explanations include Newton's Laws of Motion, particularly the second law (F = ma), which relates the forces acting on an object to its resulting acceleration. The force of gravity and the force of air resistance are the primary forces considered in these scenarios. The equation of motion under gravity (without air resistance) is given by s = ut + 0.5gt^2, where s is the displacement, u is the initial velocity, g is the acceleration due to gravity, and t is time. When air resistance is included, the motion can be modeled using more complex differential equations that account for the drag force.The final answer is: boxed{9.81}

🔑:A classic projectile motion problem! Let's break it down step by step.Step 1: Identify the given information* Initial speed (v₀) = 5.0 m/s* Angle of the roof (θ) = 30°* Distance to the ground (h) = 7 metersStep 2: Resolve the initial velocity into its componentsSince the ball rolls off the roof at an angle, we need to break down the initial velocity into its horizontal (x) and vertical (y) components.Using trigonometry, we can write:v₀x = v₀ * cos(θ) = 5.0 m/s * cos(30°) = 5.0 m/s * 0.866 = 4.33 m/s (horizontal component)v₀y = v₀ * sin(θ) = 5.0 m/s * sin(30°) = 5.0 m/s * 0.5 = 2.5 m/s (vertical component)Step 3: Determine the time of flightTo find the time of flight, we'll use the vertical component of the motion. Since the ball is under the sole influence of gravity, we can use the following kinematic equation:y = v₀y * t - (1/2) * g * t²where y is the vertical displacement (which is -h, since the ball is falling), v₀y is the initial vertical velocity, t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s²).Rearranging the equation to solve for t, we get:(1/2) * g * t² - v₀y * t - h = 0Substituting the values, we get:(1/2) * 9.8 m/s² * t² - 2.5 m/s * t - 7 m = 0Step 4: Solve the quadratic equationThis is a quadratic equation in t, which we can solve using the quadratic formula:t = (-b ± √(b² - 4ac)) / 2aIn this case, a = (1/2) * 9.8 m/s² = 4.9 m/s², b = -2.5 m/s, and c = -7 m.t = (2.5 m/s ± √((-2.5 m/s)² - 4 * 4.9 m/s² * (-7 m))) / (2 * 4.9 m/s²)t = (2.5 m/s ± √(6.25 + 137.2)) / 9.8 m/s²t = (2.5 m/s ± √143.45) / 9.8 m/s²t = (2.5 m/s ± 11.98) / 9.8 m/s²We'll only consider the positive root, since time cannot be negative:t ≈ (2.5 m/s + 11.98) / 9.8 m/s²t ≈ 14.48 / 9.8 m/s²t ≈ 1.48 secondsStep 5: Verify the resultWe can verify our result by checking if the horizontal distance traveled by the ball is consistent with the given information. Since the ball travels at a constant horizontal velocity, we can use the equation:x = v₀x * tx = 4.33 m/s * 1.48 s ≈ 6.41 metersThis is not directly relevant to the problem, but it's a good sanity check to ensure our calculation is reasonable.Therefore, the ball is in the air for approximately 1.48 seconds.

🔑:Mechanism Design: Electromagnetic Latch with Spring-Loaded SliderThe proposed mechanism utilizes an electromagnetic latch in conjunction with a spring-loaded slider to secure the drawer and allow opening only when a specific button is pressed. This design ensures the drawer remains closed under various external forces and provides a reliable and efficient solution.Components:1. Electromagnetic Latch (EL): * A solenoid coil (e.g., 12V, 1A) with a ferromagnetic core * A metal armature (e.g., steel) attached to the coil2. Spring-Loaded Slider (SLS): * A cylindrical slider (e.g., aluminum) with a diameter of 10 mm and length of 50 mm * A compression spring (e.g., stainless steel, 10 N/mm) with a free length of 20 mm and a compressed length of 10 mm3. Button and Electronics: * A push-button switch (e.g., SPST, 12V, 1A) * A control circuit (e.g., microcontroller, relay, and power supply)4. Drawer and Frame: * A standard drawer with a rectangular frame (e.g., wood or metal)Mechanism Operation:1. Closed State: The electromagnetic latch (EL) is de-energized, and the metal armature is attracted to the ferromagnetic core, holding the spring-loaded slider (SLS) in place. The compression spring is compressed, keeping the SLS engaged with the drawer frame.2. Button Pressed: When the specific button is pressed, the control circuit energizes the EL, generating a magnetic field that repels the metal armature. The armature moves away from the core, releasing the SLS.3. Slider Release: As the SLS is released, the compression spring expands, pushing the slider out of engagement with the drawer frame. The drawer is now free to open.4. Drawer Opening: The user can open the drawer by applying a force (e.g., pulling the handle).5. Drawer Closing: When the drawer is closed, the SLS is pushed back into engagement with the frame, and the compression spring compresses, holding the SLS in place.6. EL De-energization: When the button is released, the control circuit de-energizes the EL, and the metal armature returns to its original position, securing the SLS and the drawer.Calculations:1. Magnetic Force: To ensure the EL can overcome the spring force, calculate the required magnetic force:F_magnetic = F_spring * (1 + safety factor)Assuming a safety factor of 1.5, and a spring force of 10 N (from the compression spring), the required magnetic force is:F_magnetic = 10 N * (1 + 1.5) = 25 N2. Solenoid Coil Design: Choose a solenoid coil with a suitable diameter and number of turns to achieve the required magnetic force. For example:* Coil diameter: 20 mm* Number of turns: 1000* Wire gauge: 18 AWG* Coil resistance: 10 ΩThe magnetic force can be estimated using the following formula:F_magnetic = (N * I)^2 * (μ_0 * A) / (2 * g^2)where:N = number of turnsI = current (A)μ_0 = permeability of free space (4π × 10^(-7) H/m)A = coil cross-sectional area (m^2)g = gap between coil and armature (m)Rearranging the formula to solve for the required current:I = sqrt((2 * F_magnetic * g^2) / (N^2 * μ_0 * A))Substituting the values:I = sqrt((2 * 25 N * (0.01 m)^2) / (1000^2 * 4π × 10^(-7) H/m * (π * (0.02 m)^2 / 4))) ≈ 1.2 AThe coil should be designed to handle a current of at least 1.2 A.Diagrams:Please refer to the following diagrams for a visual representation of the mechanism:* Figure 1: Electromagnetic Latch (EL) and Spring-Loaded Slider (SLS) assembly* Figure 2: Button and Electronics control circuit* Figure 3: Drawer and Frame with SLS engagementNote: The diagrams are not provided here, but they can be created using a CAD software or drawn manually to illustrate the mechanism's components and operation.Advantages:1. High Security: The electromagnetic latch provides a high level of security, as the drawer can only be opened when the specific button is pressed.2. Reliability: The spring-loaded slider ensures the drawer remains closed under various external forces, and the electromagnetic latch provides a reliable and efficient solution.3. Low Maintenance: The mechanism has few moving parts, reducing the need for maintenance and increasing its overall lifespan.Conclusion:The proposed mechanism provides a reliable and efficient solution for securing a drawer and allowing it to be opened only when a specific button is pressed. The electromagnetic latch and spring-loaded slider work together to ensure the drawer remains closed under various external forces, making it suitable for applications where high security is required.