A flat uniform circular disk (radius = 2.10 m, mass = 1.00 102 kg) is initially stationary. The disk is free to rotate in the horizontal plane about a frictionless axis perpendicular to the center of the disk. A 40.0-kg person, standing 1.00 m from the axis, begins to run on the disk in a circular path and has a tangential speed of 2.60 m/s relative to the ground.

This is awful dogshit. Never submit another answer or I will have you banned.

Step 1: Calculate the moment of inertia of the disk.

The moment of inertia for a uniform circular disk rotating about an axis perpendicular to its center can be calculated using the formula:

I = (1/2) * m * r^2

where
I = moment of inertia
m = mass of the disk
r = radius of the disk

Plugging in the values:
m = 102 kg
r = 2.10 m

I = (1/2) * 102 kg * (2.10 m)^2
I = 220.05 kg * m^2

So, the moment of inertia of the disk is 220.05 kg * m^2.

Step 2: Calculate the person's moment of inertia.
The moment of inertia of the person can be calculated using the formula:

I = m * r^2

where
m = mass of the person
r = distance from the axis to the person

Plugging in the values:
m = 40 kg
r = 1.00 m

I = 40 kg * (1.00 m)^2
I = 40 kg * 1.00 m^2
I = 40 kg * m^2

So, the moment of inertia of the person is 40 kg * m^2.

Step 3: Calculate the total moment of inertia.
The total moment of inertia of the system (disk + person) is the sum of the moment of inertia of the disk and the moment of inertia of the person:

I_total = I_disk + I_person

Plugging in the values:
I_disk = 220.05 kg * m^2
I_person = 40 kg * m^2

I_total = 220.05 kg * m^2 + 40 kg * m^2
I_total = 260.05 kg * m^2

So, the total moment of inertia of the system is 260.05 kg * m^2.

Step 4: Calculate the angular velocity.
The angular velocity of the system can be calculated using the formula:

ω = v / r

where
ω = angular velocity
v = tangential speed of the person relative to the ground
r = distance from the axis to the person

Plugging in the values:
v = 2.60 m/s
r = 1.00 m

ω = 2.60 m/s / 1.00 m
ω = 2.60 rad/s

So, the angular velocity of the system is 2.60 rad/s.

To determine the angular speed of the disk after the person starts running on it, we can apply the principle of conservation of angular momentum. The total angular momentum before and after the person starts running remains constant since there are no external torques acting on the system.

The equation for angular momentum is:

L = Iω

where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Before the person starts running, the disk is initially stationary, so its initial angular momentum is zero (L_initial = 0). The person's angular momentum can be calculated as the product of their moment of inertia (m*r^2) and their angular velocity (ω_person):

L_person = m_person * r_person^2 * ω_person

where m_person is the mass of the person, r_person is the distance from the axis to the person, and ω_person is the angular velocity of the person.

After the person starts running, conservation of angular momentum tells us that the total angular momentum is still zero:

L_initial + L_person = 0

Since the initial angular momentum is zero, we can solve for the final angular velocity (ω_disk) of the entire system (disk + person) by rearranging the equation above:

L_person = -L_initial
m_person * r_person^2 * ω_person = -0

Solving for ω_person:

ω_person = -(m_person * r_person^2 * ω_person) / (I_disk + m_person * r_person^2)

To further simplify the equation, we need to calculate the moment of inertia of the disk (I_disk). For a flat uniform circular disk rotating about an axis perpendicular to its center, the moment of inertia can be found by the formula:

I_disk = (1/2) * m_disk * r_disk^2

where m_disk is the mass of the disk and r_disk is its radius.

Substituting I_disk into the equation, we get:

ω_person = -(m_person * r_person^2 * ω_person) / ((1/2) * m_disk * r_disk^2 + m_person * r_person^2)

Now, we can plug in the given values:

m_person = 40.0 kg (mass of the person)
r_person = 1.00 m (distance from the axis to the person)
ω_person = 2.60 m/s (angular velocity of the person)
m_disk = 1.00 * 10^2 kg (mass of the disk)
r_disk = 2.10 m (radius of the disk)

Plugging in these values, we can calculate the angular velocity (ω_person) of the disk.

To find Tangential spped , first you do ...

mvr=1/2MR^2w
and you're solving for w
so basically w=(2mvr)/MR^2
where m= to the mass of the disk(2.10)
r= radius of disk(102)
M=mass of person(40)
R=radius of the person(1.00)
v=tangential speed(2.60)

2(2.6)(1)(40)/(102)(2.10)=w
good luck!