Final answer:
The relative displacement of the seismic mass in an accelerometer can be calculated using the spring constant and the gravitational force component acting parallel to the spring's orientation for different tilt angles.
Step-by-step explanation:
To compute the relative displacement of the mass with respect to the housing at different tilt angles, we must consider the restoring force of the spring when the accelerometer is at an angle with respect to vertical. Because there is a damping constant involved, the system is damped, but for the purpose of this question, we're assuming maximum displacement occurs at 0° when it's perfectly oriented with the gravitational force, and to find the displacement at other angles, we need to know only the component of gravitational force acting parallel to the spring's orientation.
At 0°, which is vertical, the entire weight of the mass contributes to the stretch of the spring, since the force due to gravity (mg) acts entirely in the direction of the spring. At the equilibrium position, the spring force (kx) will equal the gravitational force (mg), i.e., kx = mg, and hence the displacement x can be found using x = mg/k. However, at angles like 30° and 60°, only a component of the gravitational force contributes to the stretch of the spring (mg*cos(θ)), hence the displacements at these angles can be found by recalculating the equilibrium position using the new force component. Simply put, the displacement decreases as the cosine of the angle increases.
For these calculations, the formula for displacement x will change with the angle. For example, at 30° the displacement x is calculated as x = (mg*cos(30°))/k, and similarly for 60°. Given the values provided: mass m = 50g (0.05kg), spring constant k = 5000 N/m, and damping constant c = 30 Ns/m, you can substitute these values into the equations to obtain the displacement at each angle.