(II) Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate ,

(b) the kinetic energy, for 𝓍 = 1.5 cm.

Verified step by step guidance

1. Step 1: Understand the system and the given values. You have a horizontal spring-mass system with a spring constant (k) of 95 N/m, a mass (m) of 75 g (which needs to be converted to kilograms), and an initial amplitude (A) of 2.0 cm (which needs to be converted to meters).
2. Step 2: Convert the mass from grams to kilograms by dividing by 1000, and convert the amplitude from centimeters to meters by dividing by 100. This will be useful for calculations involving the SI unit system.
3. Step 3: Recall the formula for the potential energy (PE) stored in a spring, which is given by PE = 0.5 * k * x^2, where x is the displacement from the equilibrium position. At maximum displacement (amplitude), the potential energy is maximum and kinetic energy is zero.
4. Step 4: To find the kinetic energy (KE) at x = 1.5 cm, first convert this displacement into meters. Then, use the conservation of mechanical energy principle, which states that the total mechanical energy (sum of potential and kinetic energy) in the system remains constant if there are no non-conservative forces like friction. Calculate the potential energy at this displacement.
5. Step 5: Since the total mechanical energy is constant, subtract the potential energy at x = 1.5 cm from the total mechanical energy (which is equal to the potential energy at the amplitude) to find the kinetic energy at x = 1.5 cm.

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 k is the spring constant and x is the displacement. This principle is fundamental in understanding how springs behave under various loads and is essential for analyzing the motion of the mass attached to the spring.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity. In the context of a mass-spring system, the kinetic energy varies as the mass oscillates, reaching its maximum when the spring is at its equilibrium position and zero at the maximum displacement.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of an object where the restoring force is proportional to the displacement from the equilibrium position. In this scenario, the mass attached to the spring undergoes SHM, characterized by a sinusoidal position-time graph, which can be used to analyze the system's energy at various points in its cycle.