A disk rolls down a slope. It rotational axis is its center of mass. It is rotation at a constant angular speed making 34.0 full revolutions in a time interval of 3.30 s.
1)What is the rotational inertia of the disk if its mass is 11.7 kg, and its radius is 49.6 cm?
A=1.44kmg^2
2) What is the rotational kinetic energy K of the rotating wheel?
3)What is the translational kinetic energy of the rolling disk?
4)What is the total kinetic energy of the rolling disk?
To find the answers to these questions, we need to use the formulas and principles related to rotational motion and kinetic energy. Let's go through them step by step:
1) What is the rotational inertia of the disk if its mass is 11.7 kg, and its radius is 49.6 cm?
The rotational inertia (also known as the moment of inertia) of a solid disk rotating about its central axis is given by the formula:
I = (1/2) * m * r^2
Where:
- I is the rotational inertia/moment of inertia.
- m is the mass of the disk.
- r is the radius of the disk.
To find the rotational inertia, plug in the given values:
I = (1/2) * 11.7 kg * (0.496 m)^2
I = 1.44 kg*m^2
So, the rotational inertia of the disk is 1.44 kg*m^2.
2) What is the rotational kinetic energy K of the rotating wheel?
The rotational kinetic energy of the disk is given by the formula:
K = (1/2) * I * ω^2
Where:
- K is the rotational kinetic energy.
- I is the rotational inertia.
- ω is the angular velocity.
In this case, the angular velocity is given as the number of revolutions per unit time (34.0 full revolutions in a time interval of 3.30 s). We need to convert it to angular velocity in radians per second:
ω = (2π * revolutions) / time
ω = (2π * 34.0) / 3.30 s
Now, we can calculate the rotational kinetic energy:
K = (1/2) * 1.44 kg*m^2 * [(2π * 34.0) / 3.30 s]^2
Simplify the expression to get the value of K.
3) What is the translational kinetic energy of the rolling disk?
When a disk rolls without slipping, it has both rotational and translational motion. The translational kinetic energy is given by:
K_translational = (1/2) * m * v^2
Where:
- K_translational is the translational kinetic energy.
- m is the mass of the disk.
- v is the linear velocity of the disk.
The linear velocity can be derived from the angular velocity using the relation v = ω * r, where r is the radius of the disk.
v = ω * r
v = [(2π * 34.0) / 3.30 s] * 0.496 m
Now, calculate the translational kinetic energy:
K_translational = (1/2) * 11.7 kg * {[(2π * 34.0) / 3.30 s] * 0.496 m}^2
Simplify the expression to get the value of K_translational.
4) What is the total kinetic energy of the rolling disk?
The total kinetic energy of the rolling disk is simply the sum of the rotational kinetic energy and the translational kinetic energy:
Total K.E. = K_rotational + K_translational
Add the previously calculated values of K_rotational and K_translational to find the total kinetic energy.