Question

Because the argument of the sine function is unitless, your calculator must be in radian mode for the rest of this question if you use it to evaluate any trigonometric functions. An object undergoing purely oscillatory motion in one dimension has an acceleration given by:

aₓ(t) = -aₘₐₓ sin(21t/T + δ).
With aₘₐₓ = 4.24 m/s², T = 9 s, and δ = 1.02 rad. What is the maximum speed of the object?

Answer 1

To find the maximum speed of an object in simple harmonic motion, calculate the angular frequency from the period and use the relation between maximum acceleration, angular frequency, and maximum velocity.

Step-by-step explanation:

Calculating Maximum Speed in Simple Harmonic Motion

To find the maximum speed of an object in simple harmonic motion, we use the equation for the velocity of the object, which is dependent on the angular frequency. The maximum velocity Vmax can be expressed as Vmax = 2πA/T, where A is the amplitude and T is the period of motion. Given that the maximum acceleration aₘₑ is 4.24 m/s² and it occurs when the object is at x = A (the maximum displacement), we can use the relationship between acceleration and velocity in simple harmonic motion to find the maximum speed.

To begin with, the maximum acceleration is related to the maximum speed and angular frequency (ω) by the equation aₘₑ = ω²A. We can find the angular frequency using the period T with the equation ω = 2π/T. In this case, the period T = 9 s, so the angular frequency ω = 2π/9 s⁻¹. Therefore, the maximum speed can be calculated by rearranging the maximum acceleration formula to Vmax = aₘₑ / ω and substituting the known values.

With these values, we can determine the maximum speed of the object undergoing oscillatory motion, which reveals the important relationship between displacement, force, and kinematics within simple harmonic motion.