A spring of negligible mass stretches 2.20 cm from its relaxed length when a force of 10.50 N is applied. A 1.100 kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x=5.0 cm and released from rest at t=0.
(c) What is the total energy of the system?
Total energy = Initial potential energy
= (1/2)kX^2
k = 10.50/0.22 = 47.7 N/m
X = initial displacement = 0.05 m
To find the total energy of the system, we need to consider both potential energy and kinetic energy.
1. Potential energy (PE):
The potential energy of a spring can be calculated using the formula:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the spring constant can be determined using Hooke's Law equation:
F = -k * x
Here, F is the force applied to the spring, which is given as 10.50 N, and x is the displacement from the relaxed length, which is 2.20 cm or 0.022 m.
So, rearranging the equation, we have:
k = -F / x
k = -10.50 N / 0.022 m
Now, we can calculate the potential energy:
PE = (1/2) * k * x^2
PE = (1/2) * (-10.50 N / 0.022 m) * (0.022 m)^2
PE = (1/2) * (-10.50 N / 0.022 m) * 0.000484 m^2
PE = -11.254 J (rounded to three decimal places)
Note that the negative sign indicates that the potential energy is stored in the system due to the compression of the spring.
2. Kinetic energy (KE):
The kinetic energy of the particle can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the particle and v is its velocity.
In this case, the mass of the particle is given as 1.100 kg.
To find the velocity, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy (sum of potential and kinetic energy) remains constant in the absence of external forces.
At the equilibrium position (x = 0), the total mechanical energy is solely potential energy since the particle is at rest and has no kinetic energy.
So,
Total energy (TE) = PE at x=0
TE = (1/2) * k * x^2
TE = (-1/2) * (10.50 N / 0.022 m) * (0.000 m)^2
TE = 0 J
At position x=5.0 cm or 0.05 m, the total energy is the sum of potential and kinetic energy.
TE = PE + KE
Since TE is constant throughout, we can write:
TE (at x=0) = TE (at x=5.0 cm)
0 J = PE (at x=5.0 cm) + KE (at x=5.0 cm)
0 J = -11.254 J (potential energy at x=5.0 cm) + KE (at x=5.0 cm)
Simplifying the equation, we can find the kinetic energy:
KE = 11.254 J (rounded to three decimal places)
Therefore, the total energy of the system is equal to the sum of the potential energy and kinetic energy:
Total energy = PE + KE
Total energy = -11.254 J + 11.254 J
Total energy = 0 J