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Analyzing Spring-Mass and Circuit Systems: Motion, Damping, and Charge Over Time

This document explores the dynamics of spring-mass systems and series circuits, focusing on motion analysis and charge behavior. It includes various scenarios: a spring with a mass of 4 kg stretched and compressed, the effect of damping, and the response of series circuits involving resistors, inductors, and capacitors. Each scenario provides a unique challenge, such as determining position over time for springs or calculating current and charge in circuits. The document combines principles of physics with mathematical methods to derive meaningful insights.

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Analyzing Spring-Mass and Circuit Systems: Motion, Damping, and Charge Over Time

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  1. 1. 2. 3. 4. A spring with a 4-kg mass has natural length 0.8 m and is maintained stretched to a length of 1.2 m by a force of 25.6 N. If the spring is compressed to a length of 0.6 m and then released with zero velocity, find the position x ( t ) of the mass at any time t. • {image} • {image} • {image} • {image}

  2. 1. 2. 3. 4. A spring with a mass of 4 kg has damping constant 32 and spring constant 128. Graph the position function of the mass at time t if it starts at the equilibrium position with a velocity of 4 m / s. • {applet} • {applet} • {applet} • {applet}

  3. 1. 2. 3. 4. Suppose a spring has mass M and spring constant k and let {image} . Suppose that the damping constant is so small that the damping force is negligible. If an external force {image} is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to find the equation that describes the motion of the mass. • {image} • {image} • {image} • {image}

  4. 1. 2. 3. 4. A series circuit consists of a resistor {image} , an inductor with L = 2 H, a capacitor with C = 0.025 F, and a 12-V battery. If the initial charge is 0.0011 C and the initial current is 0, find the current I ( t ) at time t. • {image} • {image} • {image} • {image}

  5. 1. 2. 3. 4. A series circuit consists of a resistor {image} , an inductor with L = 1 H, a capacitor with C = 0.0125 F, and a 12-V battery. If the initial charge and current are both 0, find the charge Q ( t ) at time t. • {image} • {image} • {image} • {image}

  6. 1. 2. 3. 4. A series circuit consists of a resistor {image} , an inductor with L = 1 H, a capacitor with C = 0.002 F, and a generator producing a voltage of E ( t ) = 12cos 10t. If the initial charge and current are both 0, find the charge Q ( t ) at time t. • {image} • {image} • {image} • {image}

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