A spring (k=80n/m) lies on a frictionless table, one end connected to the wall, the other to a 4kg block. The block is pulled, stretching the spring .5m. and then released.
A) max elastic potential energy of system?
b) total energy?
C) what is the acceleration of the block at the center of its vibration?
D) what is the blocks velocity when the spring is compressed .2m
To solve these questions, we need to use principles of spring-mass systems and conservation of energy. Let's go step by step:
A) To find the maximum elastic potential energy of the system, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from the equilibrium position. Mathematically, this is expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement.
In this case, the spring constant is given as k = 80 N/m, and the maximum displacement is 0.5 m. Thus, we can calculate the maximum force exerted by the spring, which is F = kx = 80 N/m * 0.5 m = 40 N.
The maximum elastic potential energy (PE) of the system is given by the equation PE = (1/2)kx^2. Plugging in the values, PE = (1/2) * 80 N/m * (0.5 m)^2 = 5 J.
Therefore, the maximum elastic potential energy of the system is 5 Joules.
B) To find the total energy of the system, we consider that at the maximum displacement, the block momentarily stops before it starts moving in the opposite direction. This implies that all the elastic potential energy has been converted to kinetic energy.
So the total energy (TE) of the system is equal to the maximum elastic potential energy at displacement. Hence, TE = PE = 5 J.
Therefore, the total energy of the system is 5 Joules.
C) The acceleration of the block at the center of its vibration can be found using the equation of motion for a mass-spring system. This equation is given as a = -(k/m) * x, where a is the acceleration, k is the spring constant, m is the mass of the block, and x is the displacement.
In this case, the spring constant is k = 80 N/m and the mass of the block is m = 4 kg. At the center of vibration, the displacement is zero (x = 0), so we have a = -(80 N/m) * 0 / 4 kg = 0 m/s^2.
Therefore, the acceleration of the block at the center of its vibration is 0 m/s^2.
D) To find the block's velocity when the spring is compressed by 0.2 m, we can use the conservation of mechanical energy. At maximum displacement, the total energy of the system is purely kinetic energy, as explained earlier.
Since the total energy (TE) of the system is 5 J, when the spring is compressed by 0.2 m, this energy will be entirely converted into the kinetic energy of the block.
The formula for kinetic energy (KE) is KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass of the block, and v is the velocity.
Therefore, we can solve for v by rearranging the equation: v = √(2KE/m).
Plugging in the values, v = √(2 * 5 J / 4 kg) = √(10/4) m/s ≈ 1.58 m/s.
Therefore, the block's velocity when the spring is compressed by 0.2 m is approximately 1.58 m/s.