Final answer:
By applying formulas for simple harmonic motion and recognizing the given motion equation as a sinusoidal function, we calculate the mass of the block, maximum speed and acceleration, and the earliest time when the block reaches a displacement of +0.1 m with a negative velocity.
Step-by-step explanation:
A simple harmonic oscillator comprising a block and a spring. To solve the problem, we use the formulas for simple harmonic motion (SHM) derived from Hooke's law and Newton's second law of motion. Given the equation of motion x=0.3sin[3πt−0.5] meters, we can identify the angular frequency ω as 3π rad/s and the amplitude A as 0.3 meters, which will help us in determining the requested quantities.
Finding the Mass of the Block (A)
From the formula ω = √(k/m), where k is the spring constant and m is the mass, we can solve for m:
m = k/ω² = 16 N/m / (3π rad/s)²
Finding the Max Speed and Acceleration (B)
The maximum speed occurs at the equilibrium position and is given by v_max = Aω = 0.3 meters * 3π rad/s. The maximum acceleration occurs at the maximum displacement and is given by a_max = Aω².
Finding the Earliest Time When x=+0.1 m with v<0 (C)
To find the time, we use the equation of motion and set x = 0.1 m and solve for t, ensuring that the derivative of the position (velocity) is negative.