Simple Energy conservation, PE = KE + PE

In summary, a block of mass M slides down a frictionless circle of radius R and flies off at an angle Theta. Using energy equations and considering the only force acting on the block is weight, the angle can be determined by setting the normal reaction N to zero and solving for theta. The centripetal force and change in potential energy can also be used to find the angle. The correct answer is inverse cosine of 1/1.5 or 48 degrees.
  • #1
Scrum
5
0

Homework Statement



A block mass M slides down the side of a frictionless circle Radius R. At an angle Theta the mass M flies off the circle, what is the angle?



Homework Equations



PE(top) = KE(point it flies off) + PE(at that point)

Arad = V^2 / R

Sum Of Forces = Mass * Acceleration

The Attempt at a Solution



Okay I actually did this one before and I was trying to do it again but somehow I don't seem to be able to get it. The answer was Inverse Cosine of 1/1.5 or 48 degrees.

The problem was done with energy equations

PE(top) = KE(point) + PE(point)

I set 0 PE to be the middle of the circle so

mgR = .5 mv^2 + mg(Rcos(theta))

mass cancels

gR = .5v^2 + gRcos(theta)



I think I'm going wrong here but I said the only force acting on the block is weight or mg, because at the point is leaves the normal force goes to 0.

so sum forces = mass * acceleration

mg = ma

mass cancels

g = a

then v^2 / R = a , so gR = V^2

then plugging that in

gR = .5v^2 + gRcos(theta)

gR = .5gR + gR cos(theta)

gR cancels

1 = .5 + cos(theta)

and I end up with 60 degrees so I think I missed out a number somewhere but I don't know where.
 
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  • #2
Scrum said:
I think I'm going wrong here but I said the only force acting on the block is weight or mg, because at the point is leaves the normal force goes to 0.

It'll leave the surface of the sphere when the normal reaction N becomes zero.

If [itex]\theta[/itex] is the angle made with the upward vertical with the radius at the position of the particle, then the centripetal force is given by,

mv^2/r = mgcos[itex]\theta[/itex] - N, which will give you mv^2 when N=0.

Also, KE = change in PE from top position, which gives,

mv^2/2 = mg(...) [you find it, in terms of r and [itex]\theta[/itex], using geometry].

From this, you'll get [itex]\theta[/itex].
 
  • #3




Your approach to this problem using energy equations is correct, but there are a couple of mistakes in your calculations. First, when setting the potential energy at the top of the circle to be zero, you need to take into account the height of the block above the center of the circle, not just the radius. This would give you an equation of mg(R+Rcos(theta)) = 0.5mv^2 + mgRcos(theta).

Secondly, when setting the sum of forces equal to ma, you need to also consider the centripetal force acting on the block, which is mv^2/R. This would give you an equation of mg = ma + mv^2/R.

Solving these equations simultaneously, you should get an angle of approximately 48 degrees, which matches the given answer.

It's important to carefully consider all forces and energies involved in a problem to ensure an accurate solution. Keep up the good work!
 

1. What is the meaning of "Simple Energy conservation"?

Simple energy conservation is the principle that states energy cannot be created or destroyed, but only transferred or converted into different forms. This means that the total amount of energy in a closed system remains constant over time.

2. What is the equation for "PE = KE + PE" and what do the variables represent?

The equation PE = KE + PE stands for potential energy equals kinetic energy plus potential energy. PE represents potential energy, which is the energy an object has due to its position or state. KE represents kinetic energy, which is the energy an object has due to its motion.

3. How does "Simple Energy conservation" apply to everyday life?

In everyday life, simple energy conservation applies to various situations such as using renewable energy sources, reducing energy consumption, and recycling materials. It also plays a role in understanding how energy is transferred and conserved in our daily activities, such as when we ride a bike or turn on a light switch.

4. Can the total amount of energy in a closed system change?

No, according to the law of conservation of energy, the total amount of energy in a closed system remains constant. Energy can only be transferred or converted between different forms, but it cannot be created or destroyed.

5. How is "Simple Energy conservation" related to the concept of work?

"Simple Energy conservation" is closely related to the concept of work, as both involve the transfer or conversion of energy. Work is defined as the transfer of energy from one system to another, and it follows the principle of energy conservation. This means that the work done on a system will result in a change in energy within that system.

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