Questions: A 0.60 kg mass is vibrating at the end of a spring. If the spring constant is 26 N / m, and the maximum displacement of the mass is 0.15 m, what is the speed of the object at its equilibrium position?

A 0.60 kg mass is vibrating at the end of a spring. If the spring constant is 26 N / m, and the maximum displacement of the mass is 0.15 m, what is the speed of the object at its equilibrium position?

Transcript text: A 0.60 kg mass is vibrating at the end of a spring. If the spring constant is $26 \mathrm{~N} / \mathrm{m}$, and the maximum displacement of the mass is 0.15 m , what is the speed of the object at its equilibrium position?

Solution

Solution Steps

Step 1: Identify the Problem

We need to find the speed of the mass at its equilibrium position when it is vibrating on a spring. The given parameters are:

Mass ($m$) = 0.60 kg
Spring constant ($k$) = 26 N/m
Maximum displacement ($x$) = 0.15 m

Step 2: Use Conservation of Energy

At the maximum displacement, all the energy is potential energy stored in the spring. At the equilibrium position, all the energy is kinetic energy. We use the conservation of mechanical energy: \[ \frac{1}{2} k x^2 = \frac{1}{2} m v^2 \]

Step 3: Solve for Velocity

Rearrange the equation to solve for velocity ($v$): \[ v = \sqrt{\frac{k}{m}} \cdot x \]

Step 4: Substitute Values

Substitute the given values into the equation: \[ v = \sqrt{\frac{26 \, \mathrm{N/m}}{0.60 \, \mathrm{kg}}} \cdot 0.15 \, \mathrm{m} \]

Step 5: Calculate the Result

Perform the calculation: \[ v = \sqrt{\frac{26}{0.60}} \cdot 0.15 \] \[ v = \sqrt{43.33} \cdot 0.15 \] \[ v \approx 6.58 \cdot 0.15 \] \[ v \approx 0.99 \, \mathrm{m/s} \]

Final Answer

$\boxed{0.99 \, \mathrm{m/s}}$

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