# Calculating Time to Reach 1/2x from Equilibrium for a Mass & Spring

• MIA6
In summary, the correct answer for the first question is 1/2x, where x is the distance the spring is stretched from its equilibrium position, and the new equilibrium position in the second question is 0.445m. The time it takes for the mass to reach the new equilibrium position would depend on the spring constant.
MIA6
1. A mass attached to the end of a spring is stretched a distance x from equilibrium and released. At what distance from equilibrium will it have acceleration equal to half its maximum acceleration?

My answer was 1/2A, but the correct answer was 1/2x, I don't know if there is a difference between these two. But I mean x may not be the amplitude, but I think the answer should be the half of its Amplitude.

2. A mass of 2.62 kg stretches a vertical spring 0.315 m. If the spring is stretched an additional 0.130 m and released, how long does it take to reach the (new) equilibrium position again?

Here, what does it mean by new equilibrium position? 0.315m is the equilibrium point?

Thanks for help.

The correct answer to your first question is 1/2x, where x is the distance the spring is stretched from its equilibrium position. The new equilibrium position in the second question would be 0.445m (the original equilibrium position plus 0.130m). To find the time it takes for the mass to reach the new equilibrium position, you would need to know the spring constant for the spring.

I would like to clarify the concept of equilibrium and its relation to the question at hand. In simple terms, equilibrium is a state where the forces acting on an object are balanced and the object is at rest. In the case of a mass attached to a spring, the equilibrium position is when the spring is neither stretched nor compressed.

Now, going back to the first question, the question is asking for the distance from equilibrium at which the mass will have acceleration equal to half its maximum acceleration. This is a different concept from the amplitude of the oscillation. The amplitude is the maximum displacement from equilibrium that the mass will reach during its oscillation, while the distance in question is the point at which the acceleration is half of its maximum value. Since acceleration is directly proportional to displacement, the answer is indeed 1/2x.

Moving on to the second question, the new equilibrium position refers to the position of the mass and spring system after the additional 0.130 m stretch. This means that the mass will now have a new equilibrium position, which is 0.315 + 0.130 = 0.445 m from its original equilibrium position. To calculate the time it takes for the mass to reach this new equilibrium position, we need to use the equation for simple harmonic motion, which is T = 2π√(m/k), where T is the period of oscillation, m is the mass, and k is the spring constant. Using this equation, we can calculate the time it takes for the mass to reach its new equilibrium position.

In conclusion, it is important to understand the concept of equilibrium and its relation to the question in order to provide the correct answer. I hope this explanation helps clarify any confusion and provides a better understanding of the topic.

## 1. How do you calculate the time to reach 1/2x from equilibrium for a mass and spring?

The time to reach 1/2x from equilibrium for a mass and spring can be calculated using the equation t = (π/2)√(m/k), where m is the mass of the object and k is the spring constant.

## 2. What is meant by "equilibrium" in this context?

In this context, equilibrium refers to the state where the spring is neither stretched nor compressed and the mass is at rest. This is also known as the resting position.

## 3. Why is it important to calculate the time to reach 1/2x from equilibrium?

Calculating the time to reach 1/2x from equilibrium is important because it helps us understand the behavior of the mass and spring system. It allows us to predict how long it will take for the system to reach a certain point, which is useful in various applications such as engineering and physics.

## 4. Can the time to reach 1/2x from equilibrium be affected by external factors?

Yes, the time to reach 1/2x from equilibrium can be affected by external factors such as air resistance and friction. These factors can slow down the movement of the mass and spring, thus increasing the time it takes to reach 1/2x from equilibrium.

## 5. What are some real-world examples of a mass and spring system?

A mass and spring system can be found in various real-world examples, such as a car's suspension system, a door with a spring hinge, a diving board, and a pogo stick. These systems use the principles of mass and spring to absorb and release energy, allowing for smooth movement and balance.

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