Conservation of Energy in circular motion

In summary, a block with mass m is initially at rest at the height of 2R on a semicircular track. It then slides without friction. The block will leave the track at a velocity of zero, and the maximum height it will reach after leaving the track can be determined by using the equation mgh = mgh' + 0.5mv'^2. The block will continue to slide until its velocity is zero, and its motion can be described using Newton's Second Law. The angle theta at the moment the block leaves the track is unknown, but the free body diagram can be drawn assuming the block is on the upper part of the track and its angle relative to an imaginary horizontal line through the center of the track
  • #36
terryds said:
As Θ increases, V increases since it is sinus function (in 0-90 degree quadrant).

Hmm.. I think I almost figured it out, right ? The only thing I miss is just the launch angle (Let's call this α )
http://www.sumoware.com/images/temp/xzihalbxxfsximxp.png [Broken]
Could you please tell me what is the relation between α and Θ ?
As theta increases, the block is rising, so must be losing KE and slowing.
Further, the fraction of the speed which is vertical is also decreasing, so it can't be as sin theta.
 
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  • #37
haruspex said:
As theta increases, the block is rising, so must be losing KE and slowing.
Further, the fraction of the speed which is vertical is also decreasing, so it can't be as sin theta.

So, it is cos theta ? Right ?
 
  • #38
terryds said:
So, it is cos theta ? Right ?
Yes vy = v cos(theta). Now, what about the value of v as a function of theta?
 
  • #39
haruspex said:
Yes vy = v cos(theta). Now, what about the value of v as a function of theta?

Do you mean the velocity as a function of theta when it's still on the track ?
E = E'
mg (2R) = mgh' + 1/2 mv^2
g (2R) = g(R+R sinΘ) + 1/2 v^2
2gR - gR - gR sinΘ = 1/2 v^2
gR (1-sin Θ) = 1/2 v^2
v = √(2 gR(1-sinΘ))

Is it right ?
 
  • #40
y
terryds said:
Do you mean the velocity as a function of theta when it's still on the track ?
E = E'
mg (2R) = mgh' + 1/2 mv^2
g (2R) = g(R+R sinΘ) + 1/2 v^2
2gR - gR - gR sinΘ = 1/2 v^2
gR (1-sin Θ) = 1/2 v^2
v = √(2 gR(1-sinΘ))

Is it right ?
Right. So vy is?
 
  • #41
haruspex said:
y
Right. So vy is?

vy = v cos Θ
vy = cos Θ √(2 gR(1-sinΘ))

The sin of angle θ is 2/3
So, the cos of angle Θ is √5 / 3
vy = √5 / 3 √(2 gR(1-(2/3)))
vy = √((10/27) gR)

Then,
Vy(t) ^2 = Vy(0) ^2 - 2 g Δy
0 = (10/27) gR - 2g Δy
Δy = 5/27 R

Then, the maximum height is R+(2/3)R+(5/27)R = (50/27) R
Right ?
Please tell me if I missed something
 
  • #42
terryds said:
vy = v cos Θ
vy = cos Θ √(2 gR(1-sinΘ))

The sin of angle θ is 2/3
So, the cos of angle Θ is √5 / 3
vy = √5 / 3 √(2 gR(1-(2/3)))
vy = √((10/27) gR)

Then,
Vy(t) ^2 = Vy(0) ^2 - 2 g Δy
0 = (10/27) gR - 2g Δy
Δy = 5/27 R

Then, the maximum height is R+(2/3)R+(5/27)R = (50/27) R
Right ?
Please tell me if I missed something
That's what I get.
 
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<h2>What is the conservation of energy in circular motion?</h2><p>The conservation of energy in circular motion is a fundamental principle in physics that states that the total energy of a system remains constant, or is conserved, as long as there are no external forces acting on the system.</p><h2>How does conservation of energy apply to circular motion?</h2><p>In circular motion, the total energy of an object is composed of its kinetic energy and potential energy. As the object moves in a circular path, the kinetic energy is constantly changing, but the total energy remains constant due to the conservation of energy principle.</p><h2>What are the types of energy involved in circular motion?</h2><p>The types of energy involved in circular motion are kinetic energy, which is the energy of motion, and potential energy, which is the energy an object has due to its position or configuration.</p><h2>What is the relationship between centripetal force and conservation of energy in circular motion?</h2><p>The centripetal force, which is the force that keeps an object moving in a circular path, is directly related to the conservation of energy in circular motion. This force is responsible for constantly changing the direction of the object's velocity, and as a result, the kinetic energy is constantly changing while the total energy remains constant.</p><h2>How does conservation of energy in circular motion affect the speed of an object?</h2><p>Conservation of energy in circular motion does not affect the speed of an object. As the object moves in a circular path, its speed may change due to changes in the direction of its velocity, but the total energy of the object remains constant.</p>

What is the conservation of energy in circular motion?

The conservation of energy in circular motion is a fundamental principle in physics that states that the total energy of a system remains constant, or is conserved, as long as there are no external forces acting on the system.

How does conservation of energy apply to circular motion?

In circular motion, the total energy of an object is composed of its kinetic energy and potential energy. As the object moves in a circular path, the kinetic energy is constantly changing, but the total energy remains constant due to the conservation of energy principle.

What are the types of energy involved in circular motion?

The types of energy involved in circular motion are kinetic energy, which is the energy of motion, and potential energy, which is the energy an object has due to its position or configuration.

What is the relationship between centripetal force and conservation of energy in circular motion?

The centripetal force, which is the force that keeps an object moving in a circular path, is directly related to the conservation of energy in circular motion. This force is responsible for constantly changing the direction of the object's velocity, and as a result, the kinetic energy is constantly changing while the total energy remains constant.

How does conservation of energy in circular motion affect the speed of an object?

Conservation of energy in circular motion does not affect the speed of an object. As the object moves in a circular path, its speed may change due to changes in the direction of its velocity, but the total energy of the object remains constant.

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