A mass of 10 kilograms is attached to the end of a spring. A force of 4 Newtons stretches the spring 10 cm. The mass is pulled 15 cm above the equilibrium point and then released with an initial downward velocity of 20 cm/sec. Determine the displacement of the mass as a function of time.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![**Problem Statement:**
A mass of 10 kilograms is attached to the end of a spring. A force of 4 Newtons stretches the spring 10 cm. The mass is pulled 15 cm above the equilibrium point and then released with an initial downward velocity of 20 cm/sec. Determine the displacement of the mass as a function of time.
**Spring Constant:**
[This section is meant for calculating or writing the spring constant, typically using Hooke's Law: \( F = k \cdot x \), where \( F \) is the force applied, \( x \) is the displacement, and \( k \) is the spring constant.]
**Differential Equation and Initial Conditions:**
[This section is intended for writing the differential equation modeling the motion of the spring-mass system and specifying initial conditions based on the problem, such as initial displacement and velocity.]
**Displacement of the Mass as a Function of Time:**
[This section should contain the solution to the differential equation, providing the mathematical expression for the displacement over time, usually involving trigonometric functions or exponentials depending on damping factors.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F1c487c1f-1731-4773-87ad-e8b7eecd803c%2Fzsdbp3_processed.jpeg&w=3840&q=75)
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