Question

A massive object of \(m=5.00 \mathrm{~kg}\) oscillates with simple harmonic motion. Its position as a function of time varies according to the equation \(x(t)=2 \sin ([\pi / 2] t+\pi / 6),\) where \(x\) is in meters. a) What are the position, velocity, and acceleration of the object at \(t=0\) s? b) What is the kinetic energy of the object as a function of time? c) At which time after \(t=0\) s is the kinetic energy first at a maximum?

Short answer

Answer: At t=0 s, the position is 1.00 m, the velocity is 2.72 m/s, and the acceleration is -2.47 m/s². The first maximum of kinetic energy occurs at t = 5/3 s.

Step-by-step solution

a) Position, velocity, and acceleration at t=0

First, let's find the position at t=0 s by substituting t=0 into our position function x(t): \(x(0) = 2\sin([ \pi / 2] \cdot 0 + \pi /6) = 2 \sin(\pi / 6) = 2 \cdot 1/2 = 1.00 \:\mathrm{m} \) Next, we need to find the velocity function by taking the first derivative of x(t) with respect to time, and then find its value at t=0 s: \(v(t) = \frac{d}{dt} 2\sin([ \pi / 2] t + \pi / 6) = \pi \cos([\pi / 2] t + \pi / 6)\) \(v(0) = \pi \cos([\pi / 2] \cdot 0 + \pi / 6) = \pi \cos(\pi / 6) = \pi \cdot (\sqrt{3}/2) \approx 2.72 \:\mathrm{m/s} \) Lastly, we need to find the acceleration function by taking the second derivative of x(t) with respect to time, and then find its value at t=0 s: \(a(t) = \frac{d^2}{dt^2} x(t) = -\frac{\pi^2}{2} \sin([\pi /2] t + \pi / 6)\) \(a(0)=-\frac{\pi^2}{2}\sin([\pi /2] \cdot 0 + \pi / 6) = -\frac{\pi^2}{4} \approx -2.47 \:\mathrm{m/s^2} \) At t=0 s, the position is 1.00 m, the velocity is 2.72 m/s, and the acceleration is -2.47 m/s².

b) Kinetic energy as a function of time

Kinetic energy (KE) can be given by the following formula: \(KE = \frac{1}{2}mv^2\) Now, we have the mass m and the velocity function v(t). So, we can plug these values into the formula: \(KE(t) = \frac{1}{2}(5.00\:\mathrm{kg})[\pi \cos([\pi / 2] t + \pi / 6)]^2\)

c) First maximum of kinetic energy

The kinetic energy will be at its maximum when the object's potential energy (PE) is at its minimum. Since the potential energy of a system in simple harmonic motion is given by \(PE(t) = \frac{1}{2}kx^2\), we need to find the first minimum of the x(t) function. This occurs when x(t) is zero because squaring zero will make the potential energy zero, so the kinetic energy will be at its maximum. We can find the first zero of x(t) by setting x(t) equal to zero and solving for time t: \(0 = 2\sin([\pi / 2] t + \pi / 6)\) \(\sin([\pi / 2] t + \pi / 6) =0\) Now we have to find the smallest positive t for which the sine is zero. The sine is zero when its argument is an integer multiple of π: \([\pi / 2] t + \pi / 6 = n\pi\), where n is an integer. Rearrange and solve for t: \(t = \frac{2}{\pi}(n\pi - \pi / 6)\) Since we want the first maximum kinetic energy, we will take the smallest positive integer value of n, which is n=1: \(t = \frac{2}{\pi}(1 \pi -\pi/6) \Rightarrow \frac{2}{\pi} \cdot \frac{5\pi}{6} = \frac{5}{3} \mathrm{s}\) Therefore, the first maximum of kinetic energy occurs at t = \(\frac{5}{3} \:\mathrm{s}\).

Key concepts

Kinetic Energy in Simple Harmonic Motion

Understanding the kinetic energy in simple harmonic motion (SHM) is fundamental in physics. Kinetic energy (KE) refers to the energy that an object possesses due to its motion. In SHM, kinetic energy changes cyclically as the object oscillates.

In our example, a mass of 5.00 kg follows SHM and the kinetic energy as a function of time is given by the equation:
\[KE(t) = \frac{1}{2} m v(t)^2\]
Where \(v(t)\) is the velocity function derived from the derivative of the position function \(x(t)\). Since the object is oscillating without external forces or friction in this ideal scenario, its total mechanical energy, which is the sum of kinetic and potential energy, remains constant.

To find the velocity, we take the derivative of the position function with respect to time. Subsequently, we can plug this velocity function into the kinetic energy equation to obtain the kinetic energy at any point in time. Importantly, at the maximum kinetic energy, the oscillating object is passing through its equilibrium position, where its potential energy is at a minimum and kinetic energy is at its peak.

Velocity and Acceleration in Simple Harmonic Motion

Velocity and acceleration are key concepts to analyze when studying SHM. Velocity is the rate of change of the object's position, while acceleration is the rate of change of velocity with time.

In the example provided, the velocity function \(v(t)\) is obtained by differentiating the position function and has the form:
\[v(t) = \pi \cos\left(\frac{\pi}{2} t + \frac{\pi}{6}\right)\]
Acceleration, on the other hand, can be determined by differentiating the velocity function, resulting in the equation:
\[a(t) = -\frac{\pi^2}{2} \sin\left(\frac{\pi}{2} t + \frac{\pi}{6}\right)\]
These derivatives are crucial for determining how fast the object is moving and how much the speed changes over time, both of which are necessary for understanding SHM in its entirety.

At any given point in the oscillatory cycle, the velocity and acceleration are out of phase; when one is maximum, the other is zero. This is a characteristic of SHM that applies to all oscillating systems.

Calculating Motion in Physics

Calculating motion involves understanding and applying the principles of kinematics and dynamics to predict the future state of a moving object. In the context of SHM, specific oscillatory motion equations help us calculate the position, velocity, acceleration, and energy of an object at any given time.

To solve for specific values, such as the position at \(t=0\) s, we directly substitute time into the given position function. Similarly, to find the velocity and acceleration at a specific time, we use the corresponding derivatives of the position function with respect to time.

In SHM, these calculated values are vital for understanding how the oscillating object behaves over time. Moreover, the calculations lead to insights on energy transformation between kinetic and potential through the oscillation cycle, which is a major aspect of energy conservation in physics.

Oscillatory Motion Equations

The equations of oscillatory motion are derived from the basic principles of SHM. These equations allow us to describe and predict the movement of oscillating systems. The general form of a position function in SHM is:
\[x(t) = A \sin(\omega t + \varphi)\]
Where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\varphi\) is the phase constant. For velocity and acceleration, the corresponding equations become the first and second derivatives of the position function.

Using these equations, students can solve various problems related to periodic motion, such as calculating period, frequency, maximum amplitude, as well as the specific position, speed, and acceleration at any instant. Mastering these equations is crucial for students to successfully analyze and comprehend the dynamics of oscillatory systems in physics.