A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's distance from equilibrium when the speed is 50 cm/s?

Verified step by step guidance

1. Step 1: Convert all given quantities into SI units. The mass of the block is 200 g = 0.2 kg, the spring constant is 10 N/m, the initial displacement is 20 cm = 0.2 m, and the initial speed is 100 cm/s = 1 m/s. The speed to analyze is 50 cm/s = 0.5 m/s.
2. Step 2: Recall the formula for the total mechanical energy in a spring-mass system: \( E = \frac{1}{2} k x^2 + \frac{1}{2} m v^2 \), where \( k \) is the spring constant, \( x \) is the displacement from equilibrium, \( m \) is the mass, and \( v \) is the speed. The total energy remains constant throughout the motion.
3. Step 3: Calculate the total mechanical energy at \( t = 0 \) using the initial conditions. Substitute \( k = 10 \; \text{N/m} \), \( x = 0.2 \; \text{m} \), \( m = 0.2 \; \text{kg} \), and \( v = 1 \; \text{m/s} \) into the energy formula: \( E = \frac{1}{2} k x^2 + \frac{1}{2} m v^2 \). This gives the total energy of the system.
4. Step 4: Use the total energy calculated in Step 3 to find the displacement \( x \) when the speed is \( v = 0.5 \; \text{m/s} \). Substitute \( v = 0.5 \; \text{m/s} \) and \( k = 10 \; \text{N/m} \) into the energy formula \( E = \frac{1}{2} k x^2 + \frac{1}{2} m v^2 \). Solve for \( x \) by isolating \( x^2 \) and taking the square root.
5. Step 5: Ensure the displacement \( x \) is expressed as the distance from the equilibrium point. Since the motion is oscillatory, the displacement can be positive or negative depending on the direction of motion, but the distance is always positive.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is essential for understanding how the spring behaves when the block is displaced and how it influences the block's motion.

Conservation of Energy

The principle of conservation of energy states that the total mechanical energy in a closed system remains constant if only conservative forces are acting. In this scenario, the potential energy stored in the spring and the kinetic energy of the block can be analyzed to determine the block's position and speed at different points in time.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion of the block attached to the spring can be modeled as SHM, characterized by a sinusoidal position and velocity over time, which is crucial for determining the block's distance from equilibrium at various speeds.