Final answer:
The distance between two adjacent nodes on the given vibrating string described by the equation y=5sin(2πx/3)cos20πt is 1.5 cm.
Step-by-step explanation:
The distance between two adjacent nodes in a vibrating string can be found by analyzing the wave equation y = 5sin(2πx/3)cos20πt. Nodes occur where the sine component equals zero because the resultant displacement y will always be zero, regardless of the value of the cosine component.
The sine component is zero when 2πx/3 is an integer multiple of π, that is, 2πx/3 = nπ where n is an integer. Solving for x gives us x = 3n/2. The distance between two adjacent nodes is the difference between two successive values of x, specifically when n increases by one. Thus the distance is 3/2 - 3(n-1)/2 = 3/2 cm. Therefore, the answer is 1.5 cm.
The equation for the standing wave is given as y(x, t) = 0.30 cm sin (3 mx + 4 st) + 0.30 cm sin (3 mx - 4 st). To write the wave function for the resulting standing wave, we can combine the two terms and rewrite it as y(x, t) = 0.30 cm cos (4 st) sin (3 mx).