A mass of 337 g connected to a light spring of force constant 19.4 N/m oscillates on a horizontal, frictionless track. The amplitude of the motion is 5.4 cm.
Calculate the total energy of the system. Answer in units of J.
To calculate the total energy of the system, we need to consider two forms of energy: potential energy (PE) and kinetic energy (KE). The total energy (E) is the sum of these two energies.
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 force constant of the spring and x is the displacement from equilibrium.
In this case, the amplitude of the motion is given as 5.4 cm, which is equal to 0.054 m.
So, PE = (1/2) * 19.4 N/m * (0.054 m)^2.
2. Kinetic Energy (KE):
The kinetic energy of the mass can be calculated using the formula: KE = (1/2) * m * v^2, where m is the mass of the object and v is the velocity of the object.
Since the motion is oscillatory, the velocity at any given point is given by v = ω * A, where ω is the angular frequency and A is the amplitude.
The angular frequency (ω) can be calculated using the formula ω = sqrt(k / m), where k is the force constant of the spring and m is the mass of the object.
So, ω = sqrt(19.4 N/m / 0.337 kg).
Next, substitute this value of ω in the formula for velocity: v = ω * A.
Finally, substitute the values for PE and KE in the total energy formula:
E = PE + KE.
Let's calculate the values step by step.
First, calculate the potential energy (PE):
PE = (1/2) * 19.4 N/m * (0.054 m)^2.
Next, calculate the angular frequency (ω):
ω = sqrt(19.4 N/m / 0.337 kg).
Then, calculate the velocity (v):
v = ω * A.
Finally, substitute the values and calculate the total energy (E):
E = PE + KE.
Make sure to use the appropriate units (meters for length, kilograms for mass, and Newtons per meter for force constant) to get the answer in joules, which is the unit for energy.