Angular Velocity after addition of mass?

In summary, the problem involves a merry-go-round with a mass of 220 kg spinning at 16 rpm. A person with a mass of 31 kg is running tangent to the merry-go-round at 5.0 m/s, jumps onto the outer edge, and the final question is what is the merry-go-round's angular velocity in rpm. The solution involves using the conservation of angular momentum and solving for the final angular velocity by setting the initial angular momentum equal to the final angular momentum. The person adds an angular momentum of mvr to the initial angular momentum of the merry-go-round, and the final system includes the merry-go-round and the person with a rotational inertia of mr^2. The correct units for angular velocity must be
  • #1
abeltyukov
32
0

Homework Statement



A merry-go-round is a common piece of playground equipment. A 4 m diameter merry-go-round with a mass of 220 kg is spinning at 16 rpm. John runs tangent to the merry-go-round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. John's mass is 31 kg. What is the merry-go-round's angular velocity, in rpm, after John jumps on?

Homework Equations



Li = Lf
KE = 1/2mv^2
KE = 1/2Iw^2

The Attempt at a Solution



I tried doing the following:

Li = Lf
(.5)(220)(2^2)(16) = (.5(220) + 31)(2^2)wf
wf = 12.482 rpm

That appears to be wrong.I also tried using:

1/2Iw^2 (KE of John and merry-go-round) = 1/2mv^2 (KE of John) + 1/2Iw^2 (KE of merry-go-round) and I also do not get the right answer.

Does something have to be converted such as 5 m/s to rpm?
Thank you!
 
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  • #2
1) ANSWER : conservation of angular momentum :

2) the initial system is the merry go round with no guy on it + the guy that is "about" to jump onto the merry go round. You have been given the intial angular mometum of the merry go round Iw and the runner gives you an angular momentum equal to : mvr (you forgot this !)

3) the final system is the merry go round + guy on it. The rotational inertia of the guy is I_runner = mr^2 (we treat him as a point particle) with r equal to the radius because the guy is sitting at the outer edge. The angular momentum of the merrry go round is Iw' and it is this w' that you need

4) Solve Iw + mvr = (I + I_runner)w' for w'

5) make sure you use THE CORRECT UNITS for w and w' (ie rad/s)

6) good luck

7) greets marlon
 
Last edited:
  • #3


I would say that your approach is on the right track, but there are a few things that need to be considered in order to get the correct answer.

Firstly, when adding mass to a rotating object, the conservation of angular momentum equation should be used, which is Iω = I'ω', where I is the moment of inertia and ω is the angular velocity before and after the addition of mass, respectively.

Secondly, the moment of inertia of a merry-go-round is not simply its mass multiplied by its radius squared, as it is a 3-dimensional object with a distributed mass. You will need to use the moment of inertia formula for a disc, which is I = 1/2mr^2.

Thirdly, when solving for the final angular velocity, make sure to convert all units to be consistent. In this case, you will need to convert the linear velocity of John (5 m/s) to angular velocity (in rpm) using the formula ω = v/r.

Once you have considered these factors, you should be able to solve for the final angular velocity of the merry-go-round after John jumps on.
 

1. What is angular velocity?

Angular velocity is a measure of the rate at which an object rotates or revolves around a fixed point. It is represented by the Greek letter omega (ω) and is measured in radians per second (rad/s).

2. How is angular velocity affected by the addition of mass?

Adding mass to an object changes its moment of inertia, which in turn affects its angular velocity. According to the Law of Conservation of Angular Momentum, if the mass is added at a distance from the axis of rotation, the angular velocity will decrease. However, if the mass is added closer to the axis of rotation, the angular velocity will increase.

3. What is the formula for calculating angular velocity?

The formula for angular velocity is ω = Δθ/Δt, where ω is angular velocity, Δθ is the change in the angle of rotation, and Δt is the change in time. This formula can also be written as ω = 2πf, where f is the frequency of rotation in hertz (Hz).

4. How is angular velocity different from linear velocity?

Angular velocity measures the rate of rotation around a fixed point, while linear velocity measures the rate of change in position of an object in a straight line. Angular velocity is also a vector quantity, meaning it has both magnitude and direction, while linear velocity is a scalar quantity with only magnitude.

5. What are some real-world applications of angular velocity?

Angular velocity is used in various fields, such as physics, engineering, and sports. It is used to calculate the rotational speed of objects, such as wheels in a car or blades in a helicopter. It is also important in understanding the motion of planets and other celestial bodies. In sports, angular velocity is used to analyze the movements of athletes, such as the rotation of a discus or the spin of a figure skater.

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