Conservation of momentum (with conservation of energy)

In summary, a 17.00kg sphere is hanging from a hook by a thin wire 3.60m long and is struck horizontally by a 6.00kg dart. The minimum initial speed of the dart needed for the combination to make a complete circular loop after the collision is calculated using the equations for centripetal force and kinetic energy. The final answer obtained for the dart's velocity may be incorrect due to a mistake in the conservation of momentum calculation.
  • #1
S[e^x]=f(u)^n
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Homework Statement


A 17.00kg sphere is hanging from a hook by a thin wire 3.60m long, it is free to swing in a complete circle. Suddenly it is struck horizontally by a 6.00kg dart that embeds itself in the sphere. What is the minimum initial speed of the dart so the combination makes a complete circular loop after the collision?

Homework Equations


[tex]F_c\mbox{top of loop} = T+m_c g\\
E_k=frac{mv^2}{2}\\
E_g=mg\Delta h[\tex]

The Attempt at a Solution


So basically what i did was figured out the velocity at the top of the loop needed to produce 0 tension since the object must make a full loop and not just get to the top.

i did this by setting the centripetal force equal to the force of gravity and the tension
since we know tension is equal to 0, the centripetal force is then equal to the gravitational force acting on the object. the masses cancel and this gives...
[tex]frac{mv^2}{r}=mg\\v=sqrt{gr}[\tex]

we know both r and g so we can then calculate the velocity at the top of the circle needed for 0 tension in the line to be 5.943m/s. Using this, we can then calculate the total energy of the particle at that point (knowing the kinetic and gravitational potential energy)
[tex]E_total = frac{mv^2}{2} + m g \delta h[\tex] (energy at the top of the loop)

this total energy if my calculations are right is equal to 2436.87 Joules. Since there's no way the system could have lost energy over the swing, i presumed that at the bottom the energy was the same(without the gravitational potential energy)... this means that the total energy must be equal to only the kinetic energy of the combined particle and dart at the bottom. so you set kinetic energy equal to 2436.87Joules and solve for the velocity of the particle at the bottom.

i know know the speed at which the particle must be traveling at the bottom. so in order to find the speed at which the dart must hit that particle (and then embed itself in) becomes a simple conservation of momentum question. where the mass of the dart and its velocity (the unknown) is the only momentum in the system prior to impact, and the only momentum after impact is the combined mass of the dart and sphere multiplied by the initial velocity needed to overcome the loop. i get an answer of 55.8m/s for the dart... but its wrong. where and how did i go wrong? sorry for the length of this post.
 
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bump :(... anybody?
 
  • #3


I would like to commend you for your detailed and organized approach to solving this problem. It is clear that you have a good understanding of the concepts involved and have applied them correctly.

However, I believe the mistake in your calculation lies in the assumption that the total energy of the system remains constant throughout the entire motion. While the conservation of energy principle holds true for isolated systems, in this case, energy is being transferred from one form to another (kinetic to potential) as the sphere swings back and forth. Therefore, the total energy of the system is not constant and cannot be used to solve for the initial velocity of the dart.

Instead, we need to consider the conservation of momentum in this system. Before the collision, the only momentum in the system is that of the dart, which can be calculated as 6.00kg * x (where x is the velocity of the dart). After the collision, the combined mass of the dart and sphere will have a momentum of (6.00kg + 17.00kg) * v (where v is the final velocity of the combined mass). Using the principle of conservation of momentum, we can equate these two values and solve for v, which will give us the final velocity of the combined mass.

Once we have the final velocity, we can use the same approach you did to calculate the initial velocity of the dart, taking into account the fact that the sphere and dart are now a combined mass. This should give you the correct answer for the minimum initial velocity of the dart.

In summary, while your approach was sound, the mistake was in using the total energy of the system instead of considering the conservation of momentum. I hope this helps clarify the issue. Keep up the good work!
 

What is the conservation of momentum?

The conservation of momentum is a fundamental law of physics that states that the total momentum of a system remains constant, unless acted upon by an external force. This means that in a closed system, the total momentum before an event is equal to the total momentum after the event.

How is conservation of momentum related to conservation of energy?

The conservation of momentum and the conservation of energy are closely related principles. Both are based on the idea that energy and momentum cannot be created or destroyed, only transferred or transformed. In a closed system, the total energy and momentum must be conserved.

What is an example of conservation of momentum in action?

A common example of conservation of momentum is a pool game. When a cue ball strikes a stationary ball, the cue ball transfers its momentum to the other ball, causing it to move. The total momentum before and after the collision remains the same, but the energy is transformed from kinetic energy of the cue ball to kinetic energy of the other ball.

Why is conservation of momentum important in science?

Conservation of momentum is important in science because it is a fundamental law that governs many physical processes. It is used to analyze and predict the motion of objects in various scenarios, such as collisions and explosions. It also helps us understand the transfer and transformation of energy in different systems.

Is conservation of momentum always true?

Yes, conservation of momentum is a universal law and has been observed to hold true in all physical systems. However, it can become more complex to apply in certain situations, such as when considering the effects of external forces or when dealing with objects at very high speeds or on a microscopic scale.

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