A 0.2 kg object that stretches a spring 6.5 cm

from its natural length when hanging at rest
oscillates with an amplitude of 1.5 cm.
The acceleration of gravity is 9.81 m/s2.
Find the total energy of the system.

The spring constant is

k = 0.2*9.81 N/0.065 m = 30.18 N/m

Total energy = (1/2) k A^2
where A is the amplitude, 0.015 m

120.3

To find the total energy of the system, we need to consider both the potential energy and the kinetic energy involved.

1. Potential Energy:
The potential energy of the spring can be calculated using Hooke's Law, which states that the potential energy stored in a spring is directly proportional to the displacement from its equilibrium position. The formula for potential energy of a spring is given by:

PE = 0.5 * k * x^2

where PE is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.

In this case, the displacement (x) is given as 6.5 cm, which is equivalent to 0.065 m. The spring constant (k) can be calculated using the formula:

k = m * g / x

where m is the mass of the object and g is the acceleration due to gravity.

Given that the mass (m) of the object is 0.2 kg and the acceleration due to gravity (g) is 9.81 m/s^2, we can calculate the spring constant (k).

k = (0.2 kg * 9.81 m/s^2) / 0.065 m

2. Kinetic Energy:
The kinetic energy of the object is given by:

KE = 0.5 * m * v^2

where KE is the kinetic energy and v is the velocity of the object.

Since the object is oscillating, we can assume that when it reaches the maximum displacement, it has maximum velocity. The maximum velocity (v) can be calculated using the formula for simple harmonic motion:

v = 2πfA

where f is the frequency of oscillation and A is the amplitude.

The frequency (f) can be calculated using the formula:

f = 1 / T

where T is the period of oscillation. For a simple harmonic motion, the period is given by:

T = 2π * √(m / k)

Substituting the values of mass (m) and spring constant (k), we can find the period (T) and then calculate the frequency (f). Finally, we can use the amplitude (A) of 1.5 cm, which is equivalent to 0.015 m, to calculate the maximum velocity (v).

3. Total Energy:
The total energy of the system is the sum of the potential energy (PE) and the kinetic energy (KE).

Total Energy = PE + KE

By plugging in the calculated values for PE and KE, we can determine the total energy of the system.