Suppose you’re eating in yet another restaurant where the dishes are shared at the table and all placed uniformly on a rotating disk-like surface. Model this surface as a thin disk of radius 30 cm. Someone else has spun the surface, such that it is initially at an angular speed of 0.6 rev/s. The surface and food has a combined mass of 3.3 kg. The waiter, to show off, throws a new dish of dumplings (mass 0.8 kg) onto the surface at a speed of 0.8 m/s, such that the dish lands on and sticks to the very edge of the surface moving in the same direction as the rotating food. While this is happening, you quickly calculate the final angular speed of the food so that you can predict its location at any time before others have a chance to eat the dumplings. What is this speed, in rad/s?
The final angular speed of the food is 1.2 rad/s.
Well, it seems like you're really spinning me around with this question! Let's break it down.
To solve this problem, we'll use the principle of conservation of angular momentum. Angular momentum is the product of moment of inertia and angular velocity.
The initial angular momentum of the system is given by the formula:
L_initial = I_initial * ω_initial
where L_initial is the initial angular momentum, I_initial is the initial moment of inertia, and ω_initial is the initial angular velocity.
The moment of inertia of a thin disk is given by the formula:
I_initial = (1/2) * m * r^2
where m is the mass of the disk and r is its radius.
For the disk and food combined, the initial moment of inertia is:
I_initial = (1/2) * (3.3 kg) * (0.3 m)^2
Now, let's calculate the initial angular momentum:
L_initial = (1/2) * (3.3 kg) * (0.3 m)^2 * (0.6 rev/s)
But we need to convert the angular velocity to rad/s:
ω_initial = (0.6 rev/s) * (2π rad/rev) = 1.2π rad/s
So, the initial angular momentum is:
L_initial = (1/2) * (3.3 kg) * (0.3 m)^2 * (1.2π rad/s)
Now, we have to take into account the new dish of dumplings. Since it sticks to the edge of the rotating surface, its moment of inertia can be neglected. Therefore, the conservation of angular momentum equation becomes:
L_initial = L_final
Where L_final is the final angular momentum.
L_final = (I_initial + I_dumplings) * ω_final
Here, I_dumplings is the moment of inertia of the dumplings. Since their mass is 0.8 kg and they are located at the very edge of the surface, their moment of inertia is:
I_dumplings = m_dumplings * r^2 = (0.8 kg) * (0.3 m)^2
Substituting this into the equation, we get:
L_final = (I_initial + (0.8 kg) * (0.3 m)^2) * ω_final
But we also know that the dumplings were thrown onto the surface at a speed of 0.8 m/s, which implies a linear momentum. However, since the linear momentum is perpendicular to the moment arm, it doesn't contribute to the angular momentum. Therefore, we can ignore it in this calculation.
Now, let's solve for ω_final:
L_initial = L_final
(1/2) * (3.3 kg) * (0.3 m)^2 * (1.2π rad/s) = ((1/2) * (3.3 kg) * (0.3 m)^2 + (0.8 kg) * (0.3 m)^2) * ω_final
Simplifying and solving for ω_final, we get:
ω_final = (1/2) * (1.2π rad/s) / ((1/2) * (1 + 0.8)) = 1.2π rad/s
So, the final angular speed of the food is approximately 1.2π rad/s.
Now, you can predict where the dumplings will end up and get ready to snatch them before anyone else does! Enjoy your meal!
To solve this problem, we can use the principle of conservation of angular momentum. The angular momentum before the dish of dumplings is thrown onto the surface is equal to the angular momentum after the dish of dumplings is added.
The formula for the angular momentum of a rotating object is:
L = Iω
where:
L is the angular momentum,
I is the moment of inertia,
ω is the angular velocity.
The moment of inertia for a thin disk is given by the formula:
I = (1/2) * m * r^2
where:
m is the mass of the object,
r is the radius of the object.
Given:
Radius of the disk (surface) = 30 cm = 0.3 m
Angular speed of the surface before the dish is thrown = 0.6 rev/s
Mass of the combined surface and food = 3.3 kg
Mass of the dish of dumplings = 0.8 kg
Speed at which the dish of dumplings is thrown = 0.8 m/s
Step 1: Calculate the initial angular momentum of the surface before the dish is thrown.
First, convert the angular speed from revolutions per second to radians per second:
ω_initial = 0.6 rev/s * 2π rad/rev = 1.2π rad/s
Next, calculate the moment of inertia of the disk:
I_initial = (1/2) * (mass of combined surface and food) * (radius)^2
= (1/2) * 3.3 kg * (0.3 m)^2
Step 2: Calculate the final angular momentum of the system after the dish of dumplings is added.
Since the dumpling dish lands and sticks to the edge of the rotating surface, its moment of inertia can be ignored because it is negligible compared to the moment of inertia of the entire system.
Therefore, the total moment of inertia after adding the dumplings is the same as the initial moment of inertia.
Step 3: Calculate the final angular speed of the system after the dish of dumplings is added.
Since the angular momentum is conserved, we can equate the initial and final angular momenta:
I_initial * ω_initial = I_final * ω_final
Solving for ω_final, we get:
ω_final = (I_initial * ω_initial) / I_final
= (I_initial * ω_initial) / I_initial
= ω_initial
= 1.2π rad/s
Therefore, the final angular speed of the food is 1.2π rad/s.
To calculate the final angular speed of the food, we can apply the principle of conservation of angular momentum. Angular momentum is given by the formula:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
The moment of inertia of the rotating surface and food can be calculated using the formula for the moment of inertia of a thin disk:
I = (1/2) * m * r^2
Where m is the mass and r is the radius of the disk.
Given that the combined mass of the surface and food is 3.3 kg and the radius is 30 cm (or 0.3 m), we can calculate the moment of inertia:
I = (1/2) * 3.3 kg * (0.3 m)^2
I = 0.1485 kg*m^2
Initially, the angular speed of the surface is given as 0.6 rev/s. Since 1 revolution is equal to 2π radians, we can convert the initial angular speed to rad/s:
ω_initial = 0.6 rev/s * 2π rad/rev
ω_initial = 1.2π rad/s
When the waiter throws the dish of dumplings onto the surface and it sticks to the edge, there is no external torque acting on the system. Therefore, the total angular momentum of the system remains constant.
Before the dumplings are thrown, the angular momentum of the system is given by:
L_initial = I * ω_initial
After the dumplings are thrown and stick to the edge of the surface, the moment of inertia of the system remains the same (as the dish is also spinning with the same angular speed). Let's denote the final angular speed of the system as ω_final.
The angular momentum after the dumplings are thrown is given by:
L_final = I * ω_final
Since angular momentum is conserved:
L_initial = L_final
I * ω_initial = I * ω_final
Simplifying the equation:
ω_final = ω_initial
ω_final = 1.2π rad/s
Therefore, the final angular speed of the food is 1.2π rad/s.