A 15 kg uniform disk of radius R = 0.25 m has a string wrapped around it, and a m = 3.3 kg weight is hanging on the string. The system of the weight and disk is released from rest.
When the 3.3 kg weight is moving with a speed of 2.1 m/s, what is the kinetic energy of the entire system?
This isn't my work, so no credit goes to me. Also, we're probably in the same class.
"Remember that a = αR, or α = a/R
Solve for acceleration by using vf2=vi2+2ax (vf=2.2, vi=0, x=(answer in part b))
That gives you the linear a… we want angular acceleration so we just divide our linear acceleration by the radius:
((2.1)^2)/(2*0.5*(2.1^2)*(3.3+(.5*15))/(3.3*9.8))/0.25
= 11.978"
No, we probably aren't in the same class. I take that back. However, my answer should still be right. It should be in rad/s^2 though: 11.978 rad/s^2
To find the kinetic energy of the entire system, we need to calculate the kinetic energy of both the weight and the disk separately, and then add them together.
First, let's calculate the kinetic energy of the weight.
The formula for kinetic energy is:
KE = 0.5 * m * v^2
Where KE is the kinetic energy, m is the mass, and v is the velocity.
Given:
m(weight) = 3.3 kg
v(weight) = 2.1 m/s
Using the formula, we can calculate the kinetic energy of the weight as follows:
KE(weight) = 0.5 * 3.3 kg * (2.1 m/s)^2
Next, let's calculate the kinetic energy of the disk.
The formula for kinetic energy of a rotating object is:
KE = 0.5 * I * ω^2
Where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
For a uniform disk rotating about its center, the moment of inertia is given by:
I = (1/2) * m * R^2
Given:
m(disk) = 15 kg
R = 0.25 m
Using the formula, we can calculate the moment of inertia of the disk as follows:
I = (1/2) * 15 kg * (0.25 m)^2
Next, we need to find the angular velocity (ω) of the disk. Since the disk is released from rest, its initial angular velocity is 0. As the weight falls, the string unwinds and the disk rotates, acquiring angular velocity.
The relationship between linear speed and angular velocity is:
v = ω * R
Where v is the linear speed, ω is the angular velocity, and R is the radius.
Given:
v(disk) = v(weight) = 2.1 m/s
R = 0.25 m
We can use the formula to find the angular velocity of the disk as follows:
v = ω * R
2.1 m/s = ω * 0.25 m
Now we can solve for ω:
ω = (2.1 m/s) / (0.25 m)
Now that we have the moment of inertia (I) and angular velocity (ω) of the disk, we can calculate its kinetic energy as follows:
KE(disk) = 0.5 * I * ω^2
Finally, to find the kinetic energy of the entire system, we add the kinetic energies of the weight and the disk:
KE(system) = KE(weight) + KE(disk)
By substituting the calculated values, we can find the answer to the question.