A 0.9 kg object that stretches a spring 2.5 cm
from its natural length when hanging at rest
oscillates with an amplitude of 8.1 cm.
Find the total energy of the system. The
acceleration due to gravity is 9.81 m/s2.
Answer in units of J
The total energy is
E = (1/2)k*X^2
where X is the oscillation amplitude, 0.081 meters.
k is the spring constant,
(0.9)*(9.81)N/0.025 m = 352 N/m
To find the total energy of the system, we need to consider both the potential energy and the kinetic energy.
The potential energy of a spring can be calculated using the formula:
PE = 1/2 * k * x^2
where PE is the potential energy, k is the spring constant, and x is the distance the spring is stretched or compressed from its natural length.
In this case, we know that the object stretches the spring 2.5 cm, so x = 0.025 m. We also know that the spring constant can be calculated using Hooke's Law:
k = F / x
where F is the force exerted by the object and x is the displacement. The force exerted by the object can be calculated using its mass and the acceleration due to gravity:
F = m * g
where m is the mass of the object and g is the acceleration due to gravity. In this case, m = 0.9 kg and g = 9.81 m/s^2.
Once we have the spring constant, we can calculate the potential energy. However, since the problem doesn't provide the spring constant, we need some additional information.
The kinetic energy of the system can be calculated using the formula:
KE = 1/2 * m * v^2
where KE is the kinetic energy, m is the mass of the object, and v is the velocity of the object.
To find the velocity, we can use the concept of conservation of energy. The total energy of the system is conserved, so at the maximum displacement (8.1 cm or 0.081 m), all the potential energy is converted into kinetic energy:
PE_max = KE_max
Substituting the formulas for potential energy and kinetic energy, we get:
1/2 * k * x^2 = 1/2 * m * v^2
Simplifying and rearranging the equation, we can solve for v:
v^2 = (k / m) * x^2
v = sqrt((k / m) * x^2)
Once we have the velocity, we can calculate the kinetic energy.
Finally, the total energy of the system is the sum of the potential energy and the kinetic energy:
Total energy = PE + KE