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 \(x(t)=2 \sin ([\pi / 2] t+\pi / 6)\). a) What is the position, velocity, and acceleration of the object at \(t=0 \mathrm{~s}^{2}\) b) What is the kinetic energy of the object as a function of time? c) At which time after \(t=0 \mathrm{~s}\) is the kinetic energy first at a maximum?

Short answer

Question: Find the position, velocity, and acceleration functions for an object undergoing simple harmonic motion, along with the position, velocity, and acceleration at t=0s. Also, find the kinetic energy function and the first time the kinetic energy reaches its maximum value after t=0s. Answer: The position function is given by: \(x(t) = 2 \sin (\frac{\pi}{2} t + \frac{\pi}{6})\) The velocity function is: \(v(t) = \pi \cos (\frac{\pi}{2} t + \frac{\pi}{6})\) The acceleration function is: \(a(t) = - \frac{\pi^2}{2} \sin (\frac{\pi}{2} t + \frac{\pi}{6})\) At t=0s: Position: \(x(0) = 2 \sin (\frac{\pi}{6}) = 1 \mathrm{~m}\) Velocity: \(v(0) = \pi \cos (\frac{\pi}{6}) = \frac{\pi \sqrt{3}}{2} \mathrm{~m/s}\) Acceleration: \(a(0) = - \frac{\pi^2}{2} \sin (\frac{\pi}{6}) = -\frac{\pi^2}{2} \mathrm{~m/s^2}\) The kinetic energy function is: \(K.E.(t) = \frac{5 \pi^2}{4} \cos^2(\frac{\pi}{2} t + \frac{\pi}{6})\) The first time the kinetic energy reaches its maximum value is at approximately \(t = 0.81\mathrm{~s}\).

Step-by-step solution

Find the velocity function and the acceleration function

Differentiate the position function \(x(t)\) with respect to time \(t\) to get the velocity function \(v(t)\), and then differentiate the velocity function to get the acceleration function \(a(t)\). \(x(t) = 2 \sin (\frac{\pi}{2} t + \frac{\pi}{6})\) \(v(t) = \frac{d}{dt}(2 \sin (\frac{\pi}{2} t + \frac{\pi}{6}))\) \(a(t) = \frac{d^2}{dt^2}(2 \sin (\frac{\pi}{2} t + \frac{\pi}{6}))\)

Calculate position, velocity, and acceleration at \(t=0 \mathrm{~s}\)

Plug in \(t=0\) into the position, velocity, and acceleration functions to find the values at \(t=0 \mathrm{~s}\). \(x(0) = 2 \sin (\frac{\pi}{2} \cdot 0 + \frac{\pi}{6})\) \(v(0) = v(0)\) (calculated in step 1) \(a(0) = a(0)\) (calculated in step 1)

Find the kinetic energy function

Using the kinetic energy formula \(K.E. = \frac{1}{2}mv^2\), plug in the mass \(m=5\mathrm{~kg}\) and the velocity function \(v(t)\) to find the kinetic energy function. \(K.E.(t) = \frac{1}{2}(5\mathrm{~kg})(v(t))^2\)

Calculate the first time the kinetic energy reaches its maximum value

To find the first time the kinetic energy is at a maximum, we need to find the critical points of the kinetic energy function by setting the derivative of the kinetic energy function with respect to time equal to zero. Solve for the time \(t\), making sure to find the first maximum after \(t=0 \mathrm{~s}\). \(\frac{d}{dt}(K.E.(t)) = 0\) Solving for \(t\) gives us the time at which the kinetic energy first reaches its maximum value.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is defined as half the product of the mass times the velocity squared. For an object of mass 'm', moving with velocity 'v', the kinetic energy 'KE' is given by the formula:\[\begin{equation} KE=\frac{1}{2}mv^2 \right).\frac{1}{2}mv^2\end{equation}\]In the context of simple harmonic motion (SHM), the velocity of the object changes constantly as it oscillates back and forth. As a result, the kinetic energy also varies with time. For a comprehensive understanding, let's consider a mass on a spring, when the mass passes through the equilibrium position, it's at maximum velocity, and consequently, it has maximum kinetic energy. Conversely, at the turning points, the velocity is zero, making the kinetic energy zero as well.

Oscillatory Motion

Oscillatory motion refers to a movement that repeatedly follows a specific path in a regular time interval, back and forth around an equilibrium position. A classic example is a pendulum or a mass-spring system engaging in simple harmonic motion, which is a type of oscillatory motion characterized by a sinusoidal wave with constant amplitude and period.

In SHM, all points along the path have a defined kinetic and potential energy, which transforms into each other while conserving the total mechanical energy of the system. An object in SHM follows a predictable restoration force proportional to its displacement from the equilibrium position, often described by Hooke's law when dealing with a spring.

Differentiation in Physics

Differentiation is a fundamental concept in calculus, often used in physics to determine how a quantity changes over time. When applied to motion, it enables us to find velocity and acceleration from the position function. Here's how it works:

• The velocity of an object is the rate of change of its position with respect to time, which we find by differentiating the position function once.
• Similarly, the acceleration is the rate of change of the velocity with respect to time, so we get it by differentiating the velocity function.

It's pivotal to understand differentiation when evaluating the behavior of systems in motion, like our SHM example, since these values are critical for establishing the kinetic energy and the nature of the motion over time.

Velocity and Acceleration

Velocity and acceleration are integral parts of analyzing motion. Velocity is a vector quantity that denotes the rate of change of an object's position, indicating both how fast it's moving and in what direction. Acceleration, also a vector quantity, communicates how the velocity of the object changes over time.

During simple harmonic motion, velocity and acceleration are at their maximums when the object crosses the equilibrium position and at their minimums (zero) at the maximum displacements. These two quantities are out of phase in SHM; when one is zero, the other is at a maximum or minimum. This relationship is pivotal as it directly influences the kinetic energy of the system, revealing the dynamic nature of objects in oscillatory motion.