Answer: The period of vibration for both vertical and horizontal motion is 0.9867 s.
Step-by-step explanation:
The period of a spring-mass oscillator can be calculated using the equation:
T = 2π * √(m/k)
where T is the period of vibration, m is the mass of the object attached to the spring, and k is the spring constant.
a) Vertical Motion:
When the system is vertical, the weight of the mass acts downward and is balanced by the upward force of the spring. The effective force acting on the mass is the force due to gravity, which is equal to the weight of the mass (mg), minus the force exerted by the spring (kx).
At the equilibrium position, the net force is zero, so we have:
mg - kx = 0
Solving for x, we get:
x = mg/k = 10 kg * 9.81 m/s^2 / 20 N/m = 4.905 m
The period of vibration is:
T = 2π * √(m/k) = 2π * √(10 kg / 20 N/m) = 0.9867 s
b) Horizontal Motion:
When the system is horizontal, the weight of the mass acts downward, but the spring force acts horizontally in the opposite direction.
At the equilibrium position, the net force is zero, so we have:
mg = kx
Solving for x, we get:
x = mg/k = 10 kg * 9.81 m/s^2 / 20 N/m = 4.905 m
The period of vibration is:
T = 2π * √(m/k) = 2π * √(10 kg / 20 N/m) = 0.9867 s
Therefore, the period of vibration for both vertical and horizontal motion is 0.9867 s.