Question

Group Activity In Exercises 33 and \(34,\) a body is moving in simple harmonic motion with position \(s=f(t)(s\) in meters, \(t\) in seconds). (a) Find the body's velocity, speed, acceleration, and jerk at time \(t\) (b) Find the body's velocity, speed, acceleration, and jerk at time \(t=\pi / 4\) sec. (c) Describe the motion of the body. $$s=2-2 \sin t$$

Short answer

The body's velocity is \(2 \cos t\), it's speed is \(2 absolute(\cos t)\), the acceleration is \(-2 sin t\) and jerk is \(-2 cos t\). At \(t = \pi / 4\) sec, the velocity is equivalent to \(\sqrt{2}\), acceleration to \(-\sqrt{2}\) and jerk to \(-\sqrt{2}\). The body is in simple harmonic motion, but the motion is dampened.

Step-by-step solution

Velocity

Firstly, find the velocity of the body at time \(t\). This is done by taking the first derivative of the position function \(s\). The derivative of \(2-2 \sin t\) is \(v(t) = 2 \cos t\) where \(v(t)\) is the velocity at time \(t\).

Speed

Speed is the absolute value of velocity. Since the value of a cosine function is always between -1 and 1, the speed is simply \(\left|2\cos t\right| = 2 \cos t\).

Acceleration

Acceleration is the rate of change of velocity, which can be found by taking the derivative of the velocity. The derivative of \(2 \cos t\) is \(a(t) = -2 \sin t\), where \(a(t)\) is the acceleration at time \(t\).

Jerk

Jerk is the rate of change of acceleration, which can be found by taking the derivative of the acceleration. The derivative of \(-2 \sin t\) is \(j(t) = -2 \cos t\), where \(j(t)\) is the jerk at time \(t\).

Substitution for \(t = \pi / 4\)

Now, substitute \(t = \pi / 4\) into the velocity, speed, acceleration, and jerk to find their values at this particular time. You get \(v(\pi/4) = 2 cos(\pi/4) = \sqrt{2}\), \(a(\pi/4) = -2 sin(\pi/4) = -\sqrt{2}\), and \(j(\pi/4) = -2 cos(\pi/4) = -\sqrt{2}\) respectively.

Description of motion

At \(t = \pi / 4\) sec, the body is moving with a speed of \(\sqrt{2}\) m/s and has an acceleration of \(-\sqrt{2}\) m/s² and a jerk of \(-\sqrt{2}\) m/s³. The negative acceleration indicates that the body is slowing down, while the negative jerk shows that this slowdown is becoming less intense over time. Therefore, the body is moving in a dampened simple harmonic motion.

Key concepts

Derivative of Trigonometric Functions

Understanding the derivative of trigonometric functions is essential for analyzing motion in physics, especially in problems dealing with simple harmonic motion (SHM). Let's delve into the reasons why. The derivative of a function at any point measures the rate at which the function's value is changing at that point.

For trigonometric functions like sine and cosine, which are often used to describe periodic motion, this translates to how fast the position of a body in SHM is changing (velocity) or how fast the velocity is changing (acceleration). For example, the derivative of the sine function is the cosine function, and vice versa, the derivative of the cosine function is the negative sine function. This relationship provides insight into how the velocity and acceleration in SHM are interconnected through these trigonometric derivatives.

Velocity in Simple Harmonic Motion

In simple harmonic motion (SHM), the velocity of an object is a vital concept since it describes how fast the object is moving and in which direction. Velocity is the first derivative of the displacement with respect to time. Considering a body in SHM represented by the position function \( s(t) = 2 - 2 \sin t \), the velocity \( v(t) \) would be \( 2 \cos t \) when derived with respect to \( t \).

This trigonometric derivative indicates that the velocity oscillates similarly to the position, reflecting the harmonic nature of the motion. The velocity is maximum when the cosine function is maximum, and minimum (in fact, zero when the motion changes direction) when the cosine function is at its minimum.

Acceleration in Simple Harmonic Motion

Acceleration is another crucial element in the study of SHM, as it measures how quickly the velocity of the object changes. For an object following SHM, acceleration is given by the second derivative of the displacement with respect to time or the first derivative of the velocity. Given the velocity \( v(t) = 2 \cos t \), the acceleration \( a(t) \) is \( -2 \sin t \) when differentiated once more with respect to \( t \).

The negative sign indicates that the acceleration acts in the opposite direction to the displacement, another defining characteristic of SHM. When the object is at the maximum displacement, it experiences zero acceleration, and when it passes through the equilibrium point, the acceleration is at its peak.

Jerk in Physics

Jerk may not be as common a term as velocity and acceleration, but it is essential in understanding the nuances of motion. Jerk is the rate of change of acceleration, or in other words, the third derivative of displacement with respect to time. It provides information regarding how the force applied on an object is changing—think of it as an 'acceleration of the acceleration'.

In the context of our SHM problem, the jerk \( j(t) \) is \( -2 \cos t \) after taking the derivative of the acceleration function \( -2 \sin t \). This shows that in SHM, jerk also oscillates with time and has similar properties as velocity. Understanding jerk in physics contextualizes the dynamics of a system subjected to periods of increasing and decreasing acceleration.

Dampened Harmonic Motion

Dampened harmonic motion describes SHM that gradually loses energy and amplitude over time due to a non-conservative force, like friction or resistance, acting against it. This energy loss modifies the behaviour of the system compared to ideal SHM, typically making the amplitude decrease exponentially with time.

In the exercise, when the jerk for \( t = \frac{\pi}{4} \) is negative, it hints at a reduction in acceleration. This could suggest the presence of damping if consistent with other motion characteristics. Even though the exercise doesn't explicitly deal with damping, the concept of dampened harmonic motion is an extension of SHM that students should be aware of, as it models real-world scenarios where energy dissipation occurs.