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Chapter 11: Problem 58

A transverse wave on a string is described by $y(x, t)=(1.2 \mathrm{cm}) \sin [(0.50 \pi \mathrm{rad} / \mathrm{s}) t-(1.00 \pi \mathrm{rad} / \mathrm{m}) x]$ Find the maximum velocity and the maximum acceleration of a point on the string. Plot graphs for displacement \(y\) versus \(t\), velocity \(v_{y}\) versus \(t\), and acceleration \(a_{y}\) versus \(t\) at \(x=0.\)

Short Answer

Expert verified
#Answer#: The maximum velocity of a point on the string is 0.6π cm/s, and the maximum acceleration is -0.3π² cm/s². At x=0: Displacement: \(y(0, t) = (1.2 \thinspace cm) \sin(0.50\pi \thinspace rad/s \cdot t)\) Velocity: \(v_y(0, t) = (1.2 \thinspace cm)(0.50\pi \thinspace rad/s) \cos(0.50\pi \thinspace rad/s \cdot t)\) Acceleration: \(a_y(0,t)=- (1.2 \thinspace cm)(0.50\pi \thinspace rad/s)^2 \sin(0.50\pi \thinspace rad/s \cdot t)\)

Step by step solution

01

1. Analyze the given equation

The transverse wave is described by the equation: \(y(x, t) = (1.2 \thinspace cm) \sin[(0.50\pi \thinspace rad/s) t - (1.00\pi \thinspace rad/m) x]\)
02

2. Find the expression for velocity

The velocity can be found by taking the partial derivative of y with respect to time t: \(v_y(x, t) = \frac{\partial y}{\partial t} = (1.2 \thinspace cm)(0.50\pi \thinspace rad/s) \cos[(0.50\pi \thinspace rad/s) t - (1.00\pi \thinspace rad/m) x]\)
03

3. Find the expression for acceleration

Now, let's find the acceleration by taking the partial derivative of v_y with respect to time t: \(a_y(x, t) = \frac{\partial v_y}{\partial t} = -(1.2 \thinspace cm)(0.50\pi \thinspace rad/s)^2 \sin[(0.50\pi \thinspace rad/s) t - (1.00\pi \thinspace rad/m) x]\)
04

4. Find the maximum velocity

The maximum value of the cosine function is 1. Therefore, the maximum velocity (v_y) at any point on the string can be found by setting the cosine component to its maximum value: \(v_{y_{max}} = (1.2 \thinspace cm)(0.50\pi \thinspace rad/s)\) \(v_{y_{max}} = 0.6\pi \thinspace cm/s\)
05

5. Find the maximum acceleration

Similarly, the maximum value of the sine function is also 1. The maximum acceleration (a_y) at any point on the string can be found by setting the sine component to its maximum value: \(a_{y_{max}} = -(1.2 \thinspace cm)(0.50\pi \thinspace rad/s)^2\) \(a_{y_{max}} = -0.3\pi^2 \thinspace cm/s^2\)
06

6. Plot the graphs at x=0

Now, let's find the expressions for displacement, velocity, and acceleration at x=0 and plot their graphs with respect to time: Displacement: \(y(0, t) = (1.2 \thinspace cm) \sin(0.50\pi \thinspace rad/s \cdot t)\) Velocity: \(v_y(0, t) = (1.2 \thinspace cm)(0.50\pi \thinspace rad/s) \cos(0.50\pi \thinspace rad/s \cdot t)\) Acceleration: \(a_y(0,t)=- (1.2 \thinspace cm)(0.50\pi \thinspace rad/s)^2 \sin(0.50\pi \thinspace rad/s \cdot t)\) Create the graphs for these expressions with the time t as the x-axis. Remember to label each graph as \(y\) versus \(t\), \(v_y\) versus \(t\), and \(a_y\) versus \(t\) respectively.

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