Conservation of energy up a ramp.

In summary, the conversation discusses a block attached to a spring and pushed to compress it a certain distance, then released to slide up a ramp to a certain maximum height. It then considers compressing the spring twice as far and the resulting maximum height. The conversation uses conservation of energy to find that the maximum height increases by a factor of 4. However, the answer from the course suggests an increase by a factor of √2, which may indicate that the original question was asking for the height to double and thus the spring factor would be √2.
  • #1
macaholic
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0

Homework Statement


Suppose there is a block attached to a spring with spring constant [itex]k[/itex]. The block is pushed so that it compresses the spring a distance [itex]x_1[/itex]. The block is released and slides without friction up a ramp, coming to a maximum height [itex]h_1[/itex] above the ground. Suppose we compressed the spring twice as far, what maximum height does the block now reach?

Homework Equations


[itex]SPE = \frac{k x^2}{2}[/itex]
[itex]GPE = m g h[/itex]

The Attempt at a Solution


Conservation of energy, for the first situation then the second:
[itex]SPE=GPE[/itex]
[itex]\frac{k x_1^2}{2} = mgh_1[/itex]
[itex] h_1=\frac{k x_1^2}{2mg}[/itex]

[itex]SPE=GPE[/itex]
[itex]\frac{k (2 x_1)^2}{2} = mgh_2[/itex]
[itex] h_2=\frac{k 4x_1^2}{2mg}[/itex]

[itex]\frac{h_2}{h_1} =\frac{k 4 x_1^2}{2mg} * \frac{2mg}{k x_1^2} = 4[/itex]

However the answerI have (this is from a course I took a long time ago) says it goes [itex]\sqrt{2}[/itex] times as high. Did I do something stupid here?
 
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  • #2
I agree with your result.
Maybe the original question was the other way round? If the height should double, which factor do you need for the spring?
 

1. How does the conservation of energy apply to an object moving up a ramp?

The law of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. This means that the total amount of energy in a closed system remains constant. When an object is moving up a ramp, its potential energy increases while its kinetic energy decreases, but the total amount of energy remains the same.

2. What factors affect the conservation of energy up a ramp?

The conservation of energy up a ramp is affected by the mass of the object, the angle of the ramp, and the presence of friction. A heavier object will require more energy to move up the ramp, and a steeper ramp will require more energy as well. Friction will also play a role in decreasing the total amount of energy available for the object to move up the ramp.

3. Can energy be lost while an object moves up a ramp?

According to the law of conservation of energy, energy cannot be lost. However, some energy may be converted into other forms, such as heat, due to the presence of friction. This means that the total amount of energy may decrease, but it is not lost.

4. What is the relationship between potential and kinetic energy when an object moves up a ramp?

As an object moves up a ramp, its potential energy increases while its kinetic energy decreases. This is because the object is gaining height, which increases its potential energy, while its speed decreases as it moves against gravity, decreasing its kinetic energy.

5. How does the conservation of energy apply to an object moving down a ramp?

The conservation of energy also applies to an object moving down a ramp. In this case, the object's potential energy decreases while its kinetic energy increases, but the total amount of energy remains constant. This is because the object is losing height, which decreases its potential energy, while gaining speed due to the force of gravity, increasing its kinetic energy.

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