A block of mass 0.25 kg is connected to a spring with spring constant 35 N/m. The block is oscillating on a frictionless horizontal surface. Its speed as it passes through its equilibrium position is 1.04 m/s. What's the total energy of the system?

The total energy is entirely kinetic energy as the block passes through the equilibrium position

E = 1/2*m*v^2

where m is the mass, and v is the velocity

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

The kinetic energy (KE) of the system can be calculated using the formula:
KE = (1/2) * m * v^2,
where m is the mass of the block and v is the speed of the block.

In this case, the mass of the block is given as 0.25 kg, and the speed of the block is given as 1.04 m/s. Plugging in these values, we have:
KE = (1/2) * 0.25 kg * (1.04 m/s)^2.

Now, since the block is connected to a spring, there is potential energy stored in the spring. The potential energy (PE) of the spring can be calculated using the formula:
PE = (1/2) * k * x^2,
where k is the spring constant and x is the displacement of the block from its equilibrium position (amplitude).

However, in this case, we are given the speed of the block (1.04 m/s) as it passes through its equilibrium position. From this information, we can determine the amplitude of the oscillation.

At the equilibrium position, the block's kinetic energy is equal to its potential energy. Therefore, we can equate the formulas for KE and PE as follows:
KE = PE.

Setting the equations equal to each other, we have:
(1/2) * 0.25 kg * (1.04 m/s)^2 = (1/2) * 35 N/m * x^2.

Simplifying the equation, we get:
0.13 kg m^2/s^2 = 17.5 N/m * x^2.

From here, we can solve for x, the amplitude of the oscillation:
x^2 = (0.13 kg m^2/s^2) / (17.5 N/m).
x^2 ≈ 0.0074285714 m^2.
x ≈ 0.08621 m.

Now we have the amplitude (x), we can calculate the potential energy of the spring:
PE = (1/2) * 35 N/m * (0.08621 m)^2.

Finally, to find the total energy of the system, we add the kinetic energy and potential energy together:
Total Energy = KE + PE.

Plugging in the values, we have:
Total Energy = (1/2) * 0.25 kg * (1.04 m/s)^2 + (1/2) * 35 N/m * (0.08621 m)^2.