Forces Acting on the Slinky

Oct 13, 2025 by miodeya.com 0 views

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Introduction

In this problem, we will explore the motion of a slinky given an initial velocity at an angle with the horizontal. We will use the principles of classical mechanics to determine the number of stairs it takes for the slinky to come to rest.

Theoretical Background

To solve this problem, we need to consider the forces acting on the slinky as it falls. The primary force acting on the slinky is gravity, which causes it to accelerate downward. However, as the slinky falls, it also experiences a force due to the tension in the slinky itself. This tension force acts in the opposite direction of the motion, causing the slinky to slow down.

Mathematical Model

We can model the motion of the slinky using the following equations:

The equation of motion for an object under the influence of gravity is given by:
$y = y_0 + v_0 \sin(\theta) t - \frac{1}{2} g t^2$
where $y$ is the height of the slinky at time $t$ , $y_0$ is the initial height, $v_0$ is the initial velocity, $\theta$ is the angle of the velocity with the horizontal, and $g$ is the acceleration due to gravity.
The equation of motion for the slinky as it falls is given by:
$x = x_0 + v_0 \cos(\theta) t$
where $x$ is the horizontal distance traveled by the slinky at time $t$ , $x_0$ is the initial horizontal position, and $v_0$ is the initial velocity.

Forces Acting on the Slinky

As the slinky falls, it experiences several forces that affect its motion. These forces include:

Gravity: The force of gravity acts downward on the slinky, causing it to accelerate downward. The magnitude of this force is given by:
$F_g = m g$
where $m$ is the mass of the slinky and $g$ is the acceleration due to gravity.
Tension: The tension in the slinky acts in the opposite direction of the motion, causing the slinky to slow down. The magnitude of this force is given by:
$F_t = k x$
where $k$ is the spring constant of the slinky and $x$ is the distance traveled by the slinky.

Energy of the Slinky

As the slinky falls, its energy changes due to the work done by the forces acting on it. The energy of the slinky can be calculated using the following equation:

Kinetic Energy: The kinetic energy of the slinky is given by:
$KE = \frac{1}{2} m v^2$
where $m$ is the mass of the slinky and $v$ is its velocity.
Potential Energy: The potential energy of the slinky is given by:
$PE = m g h$
where $m$ is the mass of the slinky, $g$ is the acceleration due to gravity, and $h$ is the height of the slinky.

Solution

To determine the number of stairs it takes for the slinky to come to rest, we need to calculate the time it takes for the slinky to fall a certain distance. We can do this by using the equation of motion for the slinky:

x = x_0 + v_0 \cos(\theta) t

We can rearrange this equation to solve for time:

t = \frac{x - x_0}{v_0 \cos(\theta)}

Once we have the time, we can calculate the number of stairs it takes for the slinky to come to rest by dividing the distance traveled by the length of each stair.

Example

Let's consider an example where the slinky is given an initial velocity of 5 m/s at an angle of 30 degrees with the horizontal. The slinky has a mass of 0.1 kg and the acceleration due to gravity is 9.8 m/s^2. The length of each stair is 0.1 m.

We can calculate the time it takes for the slinky to fall a certain distance using the equation:

t = \frac{x - x_0}{v_0 \cos(\theta)}

Plugging in the values, we get:

t = \frac{0.1 - 0}{5 \cos(30)} = 0.175 s

We can then calculate the number of stairs it takes for the slinky to come to rest by dividing the distance traveled by the length of each stair:

n = \frac{0.175}{0.1} = 1.75

Since we can't have a fraction of a stair, we round up to the nearest whole number. Therefore, it takes 2 stairs for the slinky to come to rest.

Conclusion

In this problem, we used the principles of classical mechanics to determine the number of stairs it takes for a slinky to come to rest after being given an initial velocity at an angle with the horizontal. We calculated the time it takes for the slinky to fall a certain distance and then divided the distance traveled by the length of each stair to determine the number of stairs it takes for the slinky to come to rest.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Note: The references provided are for general information purposes only and are not directly related to the specific problem discussed in this article.

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Frequently Asked Questions

Q: What is the initial velocity of the slinky?

A: The initial velocity of the slinky is given as v.

Q: What is the angle of the velocity with the horizontal?

A: The angle of the velocity with the horizontal is given as theta.

Q: What is the mass of the slinky?

A: The mass of the slinky is given as m.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is given as g.

Q: What is the length of each stair?

A: The length of each stair is given as L.

Q: How do we calculate the time it takes for the slinky to fall a certain distance?

A: We can calculate the time it takes for the slinky to fall a certain distance using the equation:

$$t = \frac{x - x_0}{v_0 \cos(\theta)}$$

Q: How do we calculate the number of stairs it takes for the slinky to come to rest?

A: We can calculate the number of stairs it takes for the slinky to come to rest by dividing the distance traveled by the length of each stair:

$$n = \frac{x}{L}$$

Q: What is the energy of the slinky?

A: The energy of the slinky can be calculated using the following equations:

*   **Kinetic Energy**: The kinetic energy of the slinky is given by:
$$KE = \frac{1}{2} m v^2$$



Potential Energy: The potential energy of the slinky is given by:
 $PE = m g h$

Q: What is the force of gravity acting on the slinky?

A: The force of gravity acting on the slinky is given by:

$$F_g = m g$$

Q: What is the tension in the slinky?

A: The tension in the slinky is given by:

$$F_t = k x$$

Q: How do we model the motion of the slinky?

A: We can model the motion of the slinky using the following equations:

*   The equation of motion for an object under the influence of gravity is given by:
$$y = y_0 + v_0 \sin(\theta) t - \frac{1}{2} g t^2$$



The equation of motion for the slinky as it falls is given by:
 $x = x_0 + v_0 \cos(\theta) t$

Common Misconceptions

Misconception 1: The slinky will always come to rest on the first stair.

A: This is not necessarily true. The slinky may take multiple stairs to come to rest, depending on the initial velocity and the angle of the velocity with the horizontal.

Misconception 2: The energy of the slinky is always conserved.

A: This is not necessarily true. The energy of the slinky may be converted from kinetic energy to potential energy as it falls, and vice versa.

Misconception 3: The force of gravity is the only force acting on the slinky.

A: This is not necessarily true. The tension in the slinky is also a force acting on the slinky, and it may be significant depending on the initial velocity and the angle of the velocity with the horizontal.

Conclusion

In this Q&A article, we have discussed the slinky given a velocity v problem and answered some of the most frequently asked questions. We have also addressed some common misconceptions about the problem and provided a better understanding of the physics involved.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Note: The references provided are for general information purposes only and are not directly related to the specific problem discussed in this article.