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We can roughly model a gymnastic tumbler as a uniform solid cylinder of mass 75 kg and diameter 1.0 m. If this tumbler rolls forward at 0.50 rev/s, (a) how much total kinetic energy does he have, and (b) what percent of his total kinetic energy is rotational?

Short Answer

Expert verified
The tumbler has 138.6 J total kinetic energy, with 33.33% being rotational.

Step by step solution

01

Calculate the Moment of Inertia

For a uniform solid cylinder, the moment of inertia I about its central axis is given by the formula I=12mr2. Here, the mass m is 75 kg, and since the diameter is 1.0 m, the radius r is 0.5 m. Plugging in these values, we get:I=12×75×(0.5)2=9.375 kg m2.
02

Determine the Angular Velocity

The angular velocity ω is related to the frequency of rotation. Given that the tumbler rolls at 0.50 revolutions per second, we convert this to radians per second (since 1 revolution is 2π radians): ω=0.50×2π=π rad/s.
03

Calculate Rotational Kinetic Energy

The rotational kinetic energy KErot is given by the formula KErot=12Iω2. Using the previously calculated I and ω:KErot=12×9.375×(π)2=46.2 J.
04

Calculate Translational Kinetic Energy

The translational kinetic energy KEtrans is given by the formula KEtrans=12mv2. The linear velocity v can be found using the relation v=rω, so v=0.5×π. Now, calculating KEtrans:KEtrans=12×75×(0.5π)2=92.4 J.
05

Calculate Total Kinetic Energy

The total kinetic energy KEtotal is the sum of the rotational and translational kinetic energies:KEtotal=KErot+KEtrans=46.2+92.4=138.6 J.
06

Calculate the Percent of Rotational Kinetic Energy

To find the percentage of the total kinetic energy that is rotational, use the formula:KErotKEtotal×100%Calculating this gives:46.2138.6×100%33.33%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia
The moment of inertia is a measure of how much an object resists changes in its rotation.Think of it as the rotational equivalent of mass for linear motion.
For a uniform solid cylinder, such as our gymnastic tumbler, the moment of inertia is calculated using the formula:
  • I=12mr2
Here:
  • m is the mass, which in this case, is 75 kg.
  • r represents the radius. Given a diameter of 1.0 m, the radius r becomes 0.5 m.
Substituting these values into the formula, we calculate:
  • I=12×75×0.52=9.375 kgm2
This value of moment of inertia tells us how much rotational force is needed for the tumble to spin about its axis.
Angular Velocity
Angular velocity describes how fast an object is rotating. In simple terms, it shows how much angle is covered per unit time.
In the exercise for the gymnastics tumbler, he rolls at a speed of 0.50 revolutions per second. But to use this value in calculations, we need to express it in radians per second because 1 revolution equals 2π radians.
The conversion is calculated as follows:
  • ω=0.50×2π=πrad/s
By converting the speed to radians per second, we can easily integrate this value into other calculations regarding rotational dynamics.
Translational Kinetic Energy
Translational kinetic energy refers to the energy of an object due to its linear motion.
For the gymnastic tumbler, you need to understand how the forward motion contributes to the overall kinetic energy. Using the relation between linear velocity and angular speed, we can explore this concept.
Firstly, to find the linear velocity v of the cylinder, use the relationship v=rω. Substituting the known values gives:
  • v=0.5×π
The translational kinetic energy can then be computed by the formula:
  • KEtrans=12mv2
Plugging in the linear velocity, we have:
  • KEtrans=12×75×(0.5π)2=92.4 J
This express the energy portion of the tumbler due to moving from one point to another in space.

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