A man stands on a platform that is rotating (without friction) with an angular speed of 1.21 rev/s; his arms are outstretched and he holds a brick in each hand The rotational inertia of the system consisting of the man, bricks, and platform about the central axis is 4.32 kg·m2. If by moving the bricks the man decreases the rotational inertia of the system to 1.13 kg·m2, (a) what is the resulting angular speed of the platform and (b) what is the ratio of the new kinetic energy of the system to the original kinetic energy?
A man stands on a frictionless rotating platform which is rotating with an angular
speed of 1.0 rev/s; his arms are outstretched and he holds a weight in each hand. With his hands in
this position the total rotational inertia of the man, the weights, and the platform is 6.0 kgm2
. If by
drawing in the weights the man decreases the rotational inertia to 2.0 kgm2
, (a) what is the resulting
angular speed of the platform? (b) By how much is the kinetic energy increased?
To solve this problem, we can apply the principle of conservation of angular momentum and the principle of conservation of kinetic energy.
(a) Angular momentum is defined as the product of rotational inertia and angular speed. According to the principle of conservation of angular momentum, the initial angular momentum of the system should be equal to the final angular momentum.
Initial angular momentum = Final angular momentum
Initial rotational inertia * Initial angular speed = Final rotational inertia * Final angular speed
(4.32 kg·m^2) * (1.21 rev/s) = (1.13 kg·m^2) * Final angular speed
Solving for Final angular speed:
Final angular speed = (4.32 kg·m^2 * 1.21 rev/s) / (1.13 kg·m^2)
Final angular speed ≈ 4.62 rev/s
Therefore, the resulting angular speed of the platform is approximately 4.62 rev/s.
(b) The ratio of the new kinetic energy of the system to the original kinetic energy can be calculated using the formula:
Ratio of new kinetic energy to original kinetic energy = (Final kinetic energy - Initial kinetic energy) / Initial kinetic energy
Both kinetic energy and rotational inertia are proportional to the square of angular speed (K.E. ∝ ω^2 and I ∝ ω^2), so the ratio simplifies to:
Ratio of new kinetic energy to original kinetic energy = (Final angular speed^2 - Initial angular speed^2) / Initial angular speed^2
Substituting the given values:
Ratio of new kinetic energy to original kinetic energy = (4.62 rev/s)^2 - (1.21 rev/s)^2 / (1.21 rev/s)^2
Calculating:
Ratio of new kinetic energy to original kinetic energy ≈ 3.23
Therefore, the ratio of the new kinetic energy of the system to the original kinetic energy is approximately 3.23.
To find the resulting angular speed of the platform, we can use the principle of conservation of angular momentum. The angular momentum of an object is given by the product of its moment of inertia and angular speed.
Angular momentum before = Angular momentum after
The initial angular momentum of the system is given by:
Angular momentum before = (Rotational inertia before) x (Angular speed before)
Angular momentum before = 4.32 kg·m² x 1.21 rev/s
Next, we have to find the new angular speed of the platform. We know that the rotational inertia of the system has decreased to 1.13 kg·m².
Angular momentum after = (Rotational inertia after) x (Angular speed after)
Angular momentum after = 1.13 kg·m² x (Angular speed after)
Since the angular momentum is conserved, we can set the two expressions for angular momentum equal to each other:
Angular momentum before = Angular momentum after
4.32 kg·m² x 1.21 rev/s = 1.13 kg·m² x (Angular speed after)
Now we can solve for the angular speed after:
Angular speed after = (4.32 kg·m² x 1.21 rev/s) / 1.13 kg·m²
Angular speed after ≈ 4.630 rev/s
So, the resulting angular speed of the platform is approximately 4.630 rev/s.
Now let's move on to part (b) to find the ratio of the new kinetic energy of the system to the original kinetic energy.
The kinetic energy of rotation is given by the equation:
Kinetic energy = (1/2) x (Rotational inertia) x (Angular speed)^2
Let's calculate the original kinetic energy:
Original kinetic energy = (1/2) x 4.32 kg·m² x (1.21 rev/s)^2
Now, let's calculate the new kinetic energy:
New kinetic energy = (1/2) x 1.13 kg·m² x (4.630 rev/s)^2
To find the ratio of the new kinetic energy to the original kinetic energy, we divide the new kinetic energy by the original kinetic energy:
Ratio = (New kinetic energy) / (Original kinetic energy)
Now you can substitute the values and calculate the ratio.