Question

A mass oscillates up and down on the end of a spring. Find its position \(s\) relative to the equilibrium position if its acceleration is \(a(t)=2 \sin t\) and its initial velocity and position are \(v(0)=3\) and \(s(0)=0,\) respectively.

Short answer

Answer: The position of the mass relative to the equilibrium position as a function of time is given by \(s(t) = -2 \sin t + 5t\).

Step-by-step solution

Integrate the acceleration function to find the velocity function

To find the velocity function, we need to integrate the acceleration function with respect to time: $$ v(t) = \int a(t) dt = \int 2 \sin t dt. $$ Integrating \(2 \sin t\) with respect to time, we obtain: $$ v(t) = -2 \cos t + C_1, $$ where \(C_1\) is the integration constant.

Apply the initial condition for velocity

We are given the initial velocity \(v(0) = 3\), so $$ v(0) = -2 \cos(0) + C_1 = 3, $$ which implies: $$ C_1 = 3+2 =5. $$ Now we have the velocity function as $$ v(t) = -2 \cos t + 5. $$

Integrate the velocity function to find the position function

To find the position function, we need to integrate the velocity function with respect to time: $$ s(t) = \int v(t) dt = \int (-2 \cos t + 5) dt. $$ Integrating \(-2 \cos t + 5\) with respect to time, we obtain: $$ s(t) = -2 \sin t + 5t + C_2, $$ where \(C_2\) is the integration constant.

Apply the initial condition for position

We are given the initial position \(s(0) = 0\), so $$ s(0) = -2 \sin(0) + 5(0) + C_2 = 0, $$ which implies: $$ C_2 = 0. $$ Now we have the position function as $$ s(t) = -2 \sin t + 5t. $$

Write the final position function

The position of the mass relative to the equilibrium position as a function of time is given by $$ s(t) = -2 \sin t + 5t. $$

Key concepts

Initial Value Problem

When dealing with differential equations, an Initial Value Problem (IVP) is a condition we apply to find a unique solution. An IVP combines a differential equation with specified values for the unknown function and its derivatives at a given point. In our example, the initial conditions are:

• Initial velocity, \( v(0) = 3 \)
• Initial position, \( s(0) = 0 \)

These conditions ensure that all the constants of integration, arising from solving the differential equations, are correctly determined. By substituting the initial velocity and position into our equations, we can solve for the integration constants \( C_1 \) and \( C_2 \). This leads us to the specific solution which describes the system's behavior over time.

Integration

Integration is a mathematical process used to find functions given their derivatives. It is the reverse operation to differentiation. In solving the motion of the mass on a spring, we use integration twice:

• First, we integrate the acceleration function \( a(t) = 2 \sin t \) to find the velocity function. The integration yields \( v(t) = -2 \cos t + C_1 \).
• Subsequently, we integrate the velocity function to obtain the position function \( s(t) = -2 \sin t + 5t + C_2 \).

Through integration, we gradually "build up" the information needed to understand how the position of the mass changes with time due to its motion.

Harmonic Motion

Harmonic motion refers to oscillatory motion that follows a repetitive and periodic path. This type of motion is prevalent in systems like mass-spring systems, pendulums, and many other physical settings. In the problem at hand, the motion of the mass is affected by its acceleration, \( a(t) = 2 \sin t \), which is a sine function. This periodic acceleration indicates that the motion will also be periodic:

• The sine function emerges in the solution, reflecting the wave-like oscillating nature.
• The position function \( s(t) = -2 \sin t + 5t \) captures this oscillation superimposed on a linear trend, due to the constant velocity component, \( 5t \).

Understanding harmonic motion is key to predicting how systems react to forces, returning to equilibrium, and displaying waves over time.