Conservation of Energy on a frictionless incline

In summary, using conservation of energy and assuming a frictionless incline, the minimum initial speed needed for a 259 g textbook to slide up a 22.1° incline that is 2.55 m long is 4.27 m/s. However, the initial speed does not match any of the given answers, suggesting a possible miscalculation or incorrect approach. Upon further examination, it was discovered that the incorrect angle of 21.5° was used in the calculations, resulting in the incorrect answer. The correct answer is 4.34 m/s.
  • #1
Ly444999
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Homework Statement


A 259 g textbook slides up a 22.1° incline that is 2.55 m long. Using conservation of energy and assuming the incline is frictionless, what minimum initial speed is needed to accomplish this?
mass = 0.259 kg
Θ = 22.1°
length of incline = 2.55 m

Homework Equations


KE = (1/2)*mv2
PE = mgh

The Attempt at a Solution


I solved for height of the incline/ramp using trigonometry where the height is opposite to the angle and the length of the ramp is the hypotenuse:
h = 2.55*sin(21.5°) = 0.934

Since the incline is frictionless the kinetic energy at the beginning is equal to the potential energy of when the textbook reaches the top of the ramp:
KE = PE
(1/2)*mv2 = mgh
Isolating for the velocity, the masses cancel out
v = √2gh
v = √(2*9.8*0.934)
= 4.27 m/s

So I got 4.27 m/s as initial velocity but it doesn't match with any of the answers which are either 4.34 m/s, 7.07 m/s, 3.07 m/s or 6.80 m/s.

So I was wondering what I did wrong. Was the velocity I was solving for not initial velocity or was the approach completely wrong? or Did I just make some miscalculations that made it not equal to the first answer?
 
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  • #2
Ly444999 said:
a 22.1° incline

Ly444999 said:
sin(21.5°)

Notice anything?
 
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  • #3
haruspex said:
Notice anything?
****, ok I'm very unintelligent, for some reason in my calculations I'm using 21.5 as my degrees I don't know why, thank you for pointing out my obvious mistake though lol.
Edit: I got right answer thank you lol.
 

1. What is the principle of conservation of energy on a frictionless incline?

The principle of conservation of energy on a frictionless incline states that the total mechanical energy of a system remains constant when an object slides down a frictionless incline. This means that the initial potential energy of the object is converted into its final kinetic energy as it moves down the incline.

2. How does the height of the incline affect the conservation of energy?

The height of the incline affects the conservation of energy as it determines the initial potential energy of the object. The higher the incline, the greater the potential energy and therefore, the greater the final kinetic energy of the object as it slides down the incline.

3. What are the factors that affect the conservation of energy on a frictionless incline?

The factors that affect the conservation of energy on a frictionless incline include the height of the incline, the mass of the object, and the angle of the incline. These factors determine the initial potential energy of the object and therefore, its final kinetic energy as it slides down the incline.

4. Does the presence of friction on the incline affect the conservation of energy?

Yes, the presence of friction on the incline can affect the conservation of energy as it can convert some of the initial potential energy of the object into heat energy. This means that the final kinetic energy of the object will be less than what it would be on a frictionless incline.

5. How is the conservation of energy on a frictionless incline related to the law of conservation of energy?

The conservation of energy on a frictionless incline is a specific application of the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed. In this case, the potential energy of the object is transformed into kinetic energy, while the total amount of energy remains constant.

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