Angular momentum and rotational energy

In summary: K1 = 1/2Iw^2 K2 = 1/2 (I+ mr^2)wnew^2 change = 1/2 (1+ mr^2)wnew^2 - 1/2Iw^2 Your change in kinetic energy : This, I'm not really sure about - I want to say that your change in kinetic energy would equal to the kinetic energy that the carousel is moving at once you sit on it .
  • #1
starstruck_
185
8

Homework Statement


A school playground has a carousel, which is simply a circular platform that can rotate around its center axis with negligible friction. This carousel has radius r=1.6 m and rotational inertia I= 177m^2kg. Suppose you are standing beside the carousel which is already spinning at w=0.47 revolutions per second, and then you suddenly sit on it, at the outer edge. Your mass is m= 98 kg.
(a) Calculate the new angular speed.
(b) Calculate the change in kinetic energy of the carousel.
(c) Calculate your change in kinetic energy.

Homework Equations



L= Iw
I = mr^2
Rotational kinetic energy: Krotational =1/2 Iw^2

The Attempt at a Solution



I know that angular momentum is conserved so I can use that to solve for the new angular velocity.

L1= L2
L1 = Iw
L2 = (I +mr^2)wnew I am using I +mr^2 for the new inertia because when you sit down, there is rotational inertia for you added onto the rotational inertia that the carousel already had. That multiplied by the new angular velocity will give you angular momentum.

Since angular momentum is conserved, L1= L2, and so, Iw = (1+ mr^2)wnew
therefore, Iw/(1_mr^2) = wnew

for kinetic energy, there is only rotational kinetic energy present since the carousel is rotating in place.

K1 = 1/2Iw^2
K2 = 1/2 (I+ mr^2)wnew^2

change = 1/2 (1+ mr^2)wnew^2 - 1/2Iw^2 Your change in kinetic energy : This, I'm not really sure about - I want to say that your change in kinetic energy would equal to the kinetic energy that the carousel is moving at once you sit on it . We do these questions online and in steps - we interpret the question ( where I determined that angular momentum is conserved) and if you don't get a step right, it won't let you go to the next step. After interpret, is develop where you develop your formulas using existing formulas - which is what I am on right now. It won't accept my answers and I'm not sure what is wrong, any help would be appreciated! Thanks :)
EDIT: I asked a friend and she ended up doing :

change = 1/2 Iwnew^2 - 1/2Iw^2

Instead of
change = 1/2 (1+ mr^2)wnew^2 - 1/2Iw^2 Not sure why though?

she used 1/2mr^2wnew^2 -didn’t add the rotational inertia of the carousel, just the rotational inertia of you.
 
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  • #2
starstruck_ said:

Homework Statement


A school playground has a carousel, which is simply a circular platform that can rotate around its center axis with negligible friction. This carousel has radius r=1.6 m and rotational inertia I= 177m^2kg. Suppose you are standing beside the carousel which is already spinning at w=0.47 revolutions per second, and then you suddenly sit on it, at the outer edge. Your mass is m= 98 kg.
(a) Calculate the new angular speed.
(b) Calculate the change in kinetic energy of the carousel.
(c) Calculate your change in kinetic energy.

Homework Equations



L= Iw
I = mr^2
Rotational kinetic energy: Krotational =1/2 Iw^2

The Attempt at a Solution



I know that angular momentum is conserved so I can use that to solve for the new angular velocity.

L1= L2
L1 = Iw
L2 = (I +mr^2)wnew I am using I +mr^2 for the new inertia because when you sit down, there is rotational inertia for you added onto the rotational inertia that the carousel already had. That multiplied by the new angular velocity will give you angular momentum.

Since angular momentum is conserved, L1= L2, and so, Iw = (1+ mr^2)wnew
therefore, Iw/(1_mr^2) = wnew
That might be correct in intent, but I'm not sure how to interpret what you wrote. You are using a '1' (as in the numeral "one") in there. Is that supposed to be the letter "I" (for moment of inertia)? Also there's an underscore that probably should be something else.

for kinetic energy, there is only rotational kinetic energy present since the carousel is rotating in place.

K1 = 1/2Iw^2
K2 = 1/2 (I+ mr^2)wnew^2

change = 1/2 (1+ mr^2)wnew^2 - 1/2Iw^2Your change in kinetic energy : This, I'm not really sure about - I want to say that your change in kinetic energy would equal to the kinetic energy that the carousel is moving at once you sit on it .We do these questions online and in steps - we interpret the question ( where I determined that angular momentum is conserved) and if you don't get a step right, it won't let you go to the next step. After interpret, is develop where you develop your formulas using existing formulas - which is what I am on right now. It won't accept my answers and I'm not sure what is wrong, any help would be appreciated! Thanks :)
EDIT: I asked a friend and she ended up doing :

change = 1/2 Iwnew^2 - 1/2Iw^2

Instead of
change = 1/2 (1+ mr^2)wnew^2 - 1/2Iw^2 Not sure why though?

she used 1/2mr^2wnew^2 -didn’t add the rotational inertia of the carousel, just the rotational inertia of you.
The problem statement (as you wrote it above), specifically part b), asks for the change of the kinetic energy of the carousel itself -- not the total change in kinetic energy.
 
  • #3
collinsmark said:
That might be correct in intent, but I'm not sure how to interpret what you wrote. You are using a '1' (as in the numeral "one") in there. Is that supposed to be the letter "I" (for moment of inertia)? Also there's an underscore that probably should be something else.The problem statement (as you wrote it above), specifically part b), asks for the change of the kinetic energy of the carousel specifically -- not the total change in kinetic energy.

Ohhhhhhhhhhhhhhhh rip my reading comprehension skills :’)

THANK YOU!
The third one makes sense now too!
 
  • Like
Likes collinsmark

Related to Angular momentum and rotational energy

What is angular momentum?

Angular momentum is a measure of an object's tendency to continue rotating at a constant rate. It is calculated as the product of an object's moment of inertia and its angular velocity.

How is angular momentum conserved?

Angular momentum is conserved in a closed system, meaning that the total angular momentum of the system remains constant over time. This means that if one part of the system gains angular momentum, another part must lose an equal amount.

What is the difference between angular momentum and linear momentum?

Angular momentum is a measure of rotational motion, while linear momentum is a measure of straight-line motion. Angular momentum is calculated using an object's moment of inertia and angular velocity, while linear momentum is calculated using an object's mass and linear velocity.

What is rotational energy?

Rotational energy is the energy an object possesses due to its rotation. It is calculated as the product of an object's moment of inertia and the square of its angular velocity.

How is rotational energy related to angular momentum?

Rotational energy and angular momentum are closely related. As an object's rotational energy increases, its angular momentum also increases. This is because an object with greater rotational energy has a higher angular velocity, thus increasing its angular momentum.

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