14-84 Kinetics of Particle: Conservation of Energy Chapter 14: Hibbeler Dynamics | Engineers Academy

Summary

This video from Engineers Academy walks through the solution of a dynamics problem involving a 4 kg smooth collar. The instructor applies the law of conservation of energy to determine the maximum distance the collar travels before it momentarily stops. The explanation covers calculating kinetic and potential energies, and using trial and error in Excel to find the precise solution. The video is an excellent resource for students learning problem-solving techniques in dynamics. Viewers are encouraged to subscribe for more problem-solving tutorials.

Takeaways

• 📚 The problem involves determining the maximum distance a 4 kg smooth collar travels before stopping momentarily.
• 🛠️ The problem is solved using the law of conservation of energy, considering both kinetic and potential energies at different points.
• 🌐 The initial conditions are a collar speed of 3 m/s at position s = 0, and the spring's unstretched length is 1 meter.
• ⚖️ Kinetic energy at point A is calculated using the formula 1/2 * mass * velocity^2, resulting in 18 joules.
• 🌀 Elastic potential energy is determined based on the spring stretch at point A, which is 0.5 meters, resulting in 12.5 joules.
• ⚖️ Gravitational potential energy is calculated using the weight of the collar and the height from a datum line, with the value 39.24 * s_max joules.
• 📊 The energy at point B is analyzed, considering that the kinetic energy is zero due to the collar's momentary stop.
• 🔗 Pythagoras' theorem is used to determine the stretch in the spring at point B, factoring in the distance s_max.
• 📈 The complex equation derived for s_max is solved using a trial-and-error method (or Excel), leading to an approximate value of 1.955 meters.
• 🎯 The final takeaway is that the maximum distance the collar travels before stopping momentarily is approximately 1.955 meters.

Q & A

What is the main problem being solved in the video?

The main problem is determining the maximum distance a 4 kg smooth collar travels before it stops momentarily, given that it has an initial speed of 3 meters per second and is connected to a spring with an unstretched length of 1 meter.

What principle is used to solve the problem?

The problem is solved using the law of conservation of energy, which states that the total kinetic and potential energy at one point is equal to the total kinetic and potential energy at another point.

How is the kinetic energy at point A calculated?

The kinetic energy at point A is calculated using the formula \( \frac{1}{2} mv^2 \), where the mass \( m \) is 4 kg and the velocity \( v \) is 3 meters per second.

What types of potential energy are considered in this problem?

Two types of potential energy are considered: elastic potential energy (due to the spring) and gravitational potential energy (due to the height of the collar).

How is the stretch in the spring at point A determined?

The stretch in the spring at point A is determined by subtracting the original length of the spring (1 meter) from the stretched length at point A (1.5 meters), resulting in a stretch of 0.5 meters.

How is the elastic potential energy at point A calculated?

The elastic potential energy at point A is calculated using the formula \( \frac{1}{2} k x^2 \), where \( k \) is the spring constant (100 N/m) and \( x \) is the stretch in the spring (0.5 meters).

What assumption is made about the gravitational potential energy at point B?

It is assumed that the gravitational potential energy at point B is zero because the datum line is chosen at point B, making the height from the datum line zero.

How is the stretch in the spring at point B determined?

The stretch in the spring at point B is determined using the Pythagorean theorem, where the stretch is the hypotenuse of a right triangle with sides equal to the maximum distance traveled and 1.5 meters.

Why is the trial and error method used to solve for the maximum distance?

The trial and error method is used because the resulting equation for the maximum distance is nonlinear and complex, making it difficult to solve algebraically.

What is the final maximum distance the collar travels before stopping?

The final maximum distance the collar travels before stopping is approximately 1.955 meters.