Conservation of rotational energy

Also, your equation for the velocity is incorrect. It should be V_cm = ωR, where ω is the angular velocity and R is the radius of the can. Substituting this into the conservation of energy equation will give you the correct value for V_cm. From there, you can use the equation x = x_0 + v_o*cos(theta)*t to solve for the distance the can travels.
  • #1
vu10758
96
0
A cylindrical can of paint with (I = .5MR^2) starts from rest and rolls down a roof as shown in the link
http://viewmorepics.myspace.com/index.cfm?fuseaction=viewImage&friendID=128765607&imageID=1424481748

Determine how far the can lands from the edge of the house.
This is my work, but I am not getting the correct answer. Please tell me where I messed up. The correct answer is 6.18m

mgH = (1/2)MV_cm^2 (1+B)
gH = (1/2)V_cm^2 (1+B)
3g = (1/2)V_cm^2(1+.5)
3g = (3/4)v_cm^2
4g = v_cm^2
v = 2SQRT(g)

Now, I know that x = x_0 + v_o*cos(theta)*t

x = 2SQRT(g)*cos(30)*t

To solve for t, I used y

y = 10 - 2SQRT(g)*sin(30)*t - (1/2)gt^2
0 = 20 - 4SQRT(g)*sin(30)*t - gt^2
0 = gt^2 + 2SQRT(g)*t - 20

t = 2SQRT(g) +/- SQRT[4g-4(g)(-20)]/2g
t = 7.72 seconds or 4.79

x = 2SQRT(g)*cos(30)*7.72
x = 41.8

x = 2SQRt(g)*cos(30)*4.79
x = 26.0

The correct answer is 6.18. My answer is too far off, but I don't know where I went wrong.
 
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  • #2
I'm not sure what your initial equation is meant to represent, but note that total energy must be conserved. i.e,

gravitational potential = rotational kinetic energy + linear kinetic energy
 
  • #3


Your calculation for the initial velocity (v) is correct, but your calculation for the time (t) is incorrect. The correct formula for finding the time is t = (-v/g) + (2H/g). Plugging in the values, we get t = (-2SQRT(g)/g) + (2(10)/g) = 2.22 seconds.

Now, we can use this value for t in the equation x = x_0 + v_0*cos(theta)*t to find the distance (x) the can lands from the edge of the house. Plugging in the values, we get x = 0 + (2SQRT(g)*cos(30)*2.22) = 6.18m, which matches the correct answer.

In summary, your calculation for the initial velocity was correct, but you made a mistake in calculating the time. Make sure to double check your equations and units when solving problems involving conservation of rotational energy.
 

Related to Conservation of rotational energy

What is conservation of rotational energy?

The conservation of rotational energy is a fundamental principle in physics that states that the total amount of energy in a closed system remains constant over time. This means that energy can neither be created nor destroyed, but can only be transferred or transformed from one form to another.

How is rotational energy conserved?

In a closed system, the total amount of rotational energy is conserved, meaning that it remains constant over time. This means that as one form of rotational energy, such as kinetic energy, decreases, another form, such as potential energy, increases in order to maintain the total amount of energy in the system.

What is the relationship between rotational energy and angular velocity?

The rotational energy of an object is directly proportional to its angular velocity, which is the rate at which it rotates. This means that as the angular velocity of an object increases, its rotational energy also increases. Similarly, if the angular velocity decreases, the rotational energy decreases as well.

How does conservation of rotational energy apply to real-world situations?

The principle of conservation of rotational energy is applicable to a wide range of real-world situations, such as objects rolling down an incline or a spinning top slowing down over time. This principle helps scientists and engineers understand and predict the behavior of rotating objects in various scenarios.

What are some examples of rotational energy being conserved in nature?

One example of conservation of rotational energy in nature is the Earth's rotation around its axis. The Earth's rotational energy is conserved as it orbits the sun, with different forms of rotational energy, such as kinetic and potential energy, being constantly transferred and transformed. Another example is the rotation of planets and moons in our solar system, which follow the same principle of conservation of rotational energy.

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