A 2 kg mass is hanging from a spring with a constant of 1500 N/m. The mass is pulled down a distance of 0.3 m and then released. If the specific heat capacity of the spring is 0.35 J/gK, the spring has a mass of 10 g and the initial temperature of the spring is 20 degrees Celsius, what will be the temperature of the spring once it is done bouncing?

Question

A 2 kg mass is hanging from a spring with a constant of 1500 N/m. The mass is pulled down a distance of 0.3 m and then released. If the specific heat capacity of the spring is 0.35 J/gK, the spring has a mass of 10 g and the initial temperature of the spring is 20 degrees Celsius, what will be the temperature of the spring once it is done bouncing?

Answer 1

To calculate the final temperature of the spring, we need to consider the energy transferred to the spring during its bouncing motion.

First, let's determine the initial potential energy of the mass-spring system when it is pulled down by 0.3 m. The potential energy can be calculated using the formula:

Potential Energy (PE) = (mass) * (gravity) * (height)

Here, the mass is given as 2 kg and the height is 0.3 m. The acceleration due to gravity is approximately 9.8 m/s^2. Substituting these values into the formula, we get:

PE = 2 kg * 9.8 m/s^2 * 0.3 m = 5.88 J

Next, let's find the maximum velocity of the mass when it reaches its equilibrium point. The potential energy is entirely converted into kinetic energy at this point.

The formula to calculate kinetic energy is:

Kinetic Energy (KE) = 0.5 * (mass) * (velocity)^2

Considering that the mass is 2 kg, the kinetic energy will be equal to the potential energy, i.e., 5.88 J.

5.88 J = 0.5 * 2 kg * (velocity)^2

Simplifying the equation:

2.94 J = (velocity)^2

Taking the square root:

velocity = sqrt(2.94 J) ≈ 1.71 m/s

Now, let's determine the change in temperature of the spring. The formula to calculate the heat transfer is:

Heat (Q) = (mass) * (specific heat capacity) * (change in temperature)

Here, the mass of the spring is given as 10 g, which is equivalent to 0.01 kg. The specific heat capacity is provided as 0.35 J/gK. We need to convert the mass to grams, so we have:

Q = 0.01 kg * (0.35 J/gK) * (change in temperature)

Since the initial temperature of the spring is given as 20°C, the change in temperature is the difference between the final temperature (Tf) and the initial temperature (Ti).

Q = 0.01 kg * (0.35 J/gK) * (Tf - 20)

Now, let's substitute the calculated values:

5.88 J = 0.01 kg * (0.35 J/gK) * (Tf - 20)

Simplifying the equation, we get:

Tf - 20 = (5.88 J) / (0.01 kg * (0.35 J/gK))

Tf - 20 ≈ 168

Finally, solving for Tf:

Tf ≈ 188°C

Therefore, the temperature of the spring once it is done bouncing will be approximately 188°C.