To find the amplitudes of displacement and velocity for a body undergoing sinusoidal acceleration, one must use the angular frequency and understand the relationship between maximum displacement, stiffness, and mass. The maximum velocity is directly proportional to the amplitude and angular frequency.
The question asks us to find the amplitudes of displacement and velocity of a body experiencing sinusoidal acceleration at a frequency of 8 Hz, with a given maximum acceleration. The maximum displacement amplitude is also known as X, and the maximum velocity amplitude is represented as Umax. The angular frequency (ω) is essential for finding these amplitudes.
Given the frequency (f) is 8 Hz, the angular frequency can be calculated using ω = 2πf. For sinusoidal motion in simple harmonic motion (SHM), the displacement as a function of time is x(t) = X cos(2πft), and the maximum acceleration is given as amax = Xω². The velocity is a sinusoidal function as well, leading to the conclusion that Umax = Xω. However, without the maximum displacement value, we cannot compute the exact amplitudes.
Overall, the key concepts involve understanding the relationship between maximum acceleration, maximum velocity, and maximum displacement in SHM. The maximum velocity depends on amplitude, the stiffness of the system (force constant k), and the mass of the object (m). The greater the amplitude and stiffness, the greater the maximum velocity, but it is smaller for objects with larger masses.