Conservation of energy problem

In summary, the minimum velocity needed for a mass attached to a pendulum of length L, at an initial angle theta (8) and velocity Vo, to go from its lowest point to its highest point is the square root of 3gL + 2gLcos(8). This is due to the fact that the mass must maintain enough kinetic energy to travel on a circular path and reach the highest point, in addition to the potential energy required to reach the lowest point. Neglecting the kinetic energy term in the equation will result in an incorrect solution.
  • #1
OVB
32
0
A mass is attached to a pendulum of length L that is at an angle theta (8) with respect to the normal. The mass has an initial velocity Vo. What is the minimum speed (symbolically) needed for the mass to go all the way down to its lowest point and all the way back up its highest point (as in, the rope of the pendulum is projected vertically upwards).

I set 1/2(m)Vo^2 + mgL(1-cos(8)) = 2mgL

and got Vo = square root of 2gL + 2gLcos(8), but my book says it is the square root of 3gL + 2gLcos(8). I don't see what's wrong with my set up, as I have checked the result of this set up several times and got the same answer.
 
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  • #2
no, I meant that it was originally at an angle with respect to what it would be at rest (i.e., bob below pivot) , and the velocity is supposed to be like that since it is supposed to be solved symbolically.
 
  • #3
OVB said:
no, I meant that it was originally at an angle with respect to what it would be at rest (i.e., bob below pivot) , and the velocity is supposed to be like that since it is supposed to be solved symbolically.
OOPs. I inadvertantly deleted my first message. And OOPs again, I did misinterpret the potential energy term in your original equation. You are quite correct about the initial potential energy. However, I was correct that you have neglected the velocity that the bob must have at the higest point. The bob cannot possibly reach that point with zero velocity. In order to reach that point, the bob must travel on a circular path. The net force at the top of the circle must be just enough to keep it in circular motion. If the bob had zero kinetic energy at the top, it would fall straight down, but it could never even get to the top if it didn't have enough kinetic energy in the first place. You need a kinetic energy term on the right hand side of your equation, and you need to find the velocity from circular motion considerations.
 

Related to Conservation of energy problem

1. What is the law of conservation of energy?

The law of conservation of energy states that energy cannot be created or destroyed, but it can be transferred from one form to another.

2. Why is conservation of energy important?

Conservation of energy is important because it helps us understand and predict how energy behaves in different systems. It also allows us to use energy more efficiently and reduce waste.

3. How is energy conserved in a closed system?

In a closed system, the total amount of energy remains constant. This means that energy cannot enter or leave the system, but it can be converted from one form to another. Therefore, the energy within the system is conserved.

4. Can energy be completely conserved?

No, energy cannot be completely conserved. In any energy transfer or transformation, some energy is always lost as heat or other forms of energy. This is known as energy dissipation or degradation.

5. How does conservation of energy relate to the environment?

The law of conservation of energy is a fundamental principle in environmental science. It helps us understand the flow of energy in ecosystems and the impact of human activities on the environment. By conserving energy, we can reduce our impact on the environment and promote sustainability.

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