A block rides on a piston that is moving vertically with simple harmonic motion.
(a) If the SHM has period 3.1 s, at what amplitude of motion will the block and piston separate?
(b) If the piston has an amplitude of 9.6 cm, what is the maximum frequency for which the block and piston will be in contact continuously?
To answer these questions, we need to understand the relationship between the simple harmonic motion of the piston and the separation of the block.
(a) To determine the amplitude of motion at which the block and piston will separate, we need to consider the maximum displacement of the piston during its oscillation. At maximum displacement, the acceleration of the piston becomes zero, and therefore the gravitational force acting on the block will exceed the maximum force exerted by the piston. This will cause the block to separate from the piston.
To find this maximum displacement, we can use the concept of equilibrium of forces. At maximum displacement, the gravitational force acting on the block will be equal to the maximum force exerted by the piston. The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity.
The maximum force exerted by the piston can be calculated using the formula for force in simple harmonic motion:
F_piston = k * A
where k is the spring constant of the piston and A is the amplitude of motion.
Setting F_gravity equal to F_piston, we can solve for the amplitude A:
m * g = k * A
A = (m * g) / k
Therefore, the amplitude of motion at which the block and piston will separate is given by A = (m * g) / k, where m is the mass of the block, g is the acceleration due to gravity, and k is the spring constant of the piston.
(b) To determine the maximum frequency for which the block and piston will be in contact continuously, we need to consider the maximum displacement of the piston and the speed at which it oscillates.
The maximum frequency occurs when the piston reaches its maximum displacement and changes its direction of motion. At this point, the piston's velocity is zero. If the frequency is any higher than this maximum frequency, the piston will not have enough time to return to its original position before changing direction, causing the block to separate from the piston.
The maximum frequency can be calculated using the formula:
f_max = (1 / T_min)
where T_min is the minimum period required for the piston to complete one full oscillation (from maximum displacement to maximum displacement).
The minimum period can be calculated using the formula for period in simple harmonic motion:
T_min = 2π * sqrt(m / k)
where m is the mass of the block and k is the spring constant of the piston.
Therefore, the maximum frequency for which the block and piston will be in contact continuously is given by f_max = (1 / T_min), where T_min = 2π * sqrt(m / k), m is the mass of the block, and k is the spring constant of the piston.