A lazy Susan consists of a heavy plastic disk mounted on a frictionless bearing resting on a vertical shaft through its center. The cylinder has a radius R = 10 cm and mass M = 0.24 kg. A cockroach (mass m = 0.015 kg) is on the lazy Susan, at a distance of 10 cm from the center. Both the cockroach and the lazy Susan are initially at rest. The &roach then walks along a circular path concentric with the axis of the lazy Susan at a constant distance of 10 cm from the axis of the shaft. If the speed of the cockroach with respect to the lazy Susan is 0.01 m/s, what is the speed of the cockroach with respect to the room?

To find the speed of the cockroach with respect to the room, we need to consider the velocities of both the cockroach and the lazy Susan and apply the principle of relative motion.

Let's break down the problem into steps:

Step 1: Find the initial angular velocity of the lazy Susan.
The lazy Susan and the cockroach initially are at rest. Therefore, the initial angular velocity of the lazy Susan is zero.

Step 2: Find the final angular velocity of the lazy Susan.
The cockroach walks along a circular path concentric with the axis of the lazy Susan. As the cockroach moves, the lazy Susan experiences an equal and opposite angular displacement to keep the system in equilibrium. The cockroach completes one full revolution, so it moves a distance equal to the circumference of the circle it is walking on.

The circumference of the circle is given by:
C = 2πr = 2π(0.10 m) = 0.628 m

Since the time taken to complete the circular path (one revolution) is the same for both the cockroach and the lazy Susan, the final angular displacement of the lazy Susan is also equal to one full revolution.

Therefore, the final angular velocity (ω) of the lazy Susan is given by:
ω = Δθ / Δt,
where Δθ is the angular displacement and Δt is the time taken.

Since the angular displacement is 2π (one full revolution), and the time taken is the same for the cockroach (Δt = 2πr / v, where v is the velocity of the cockroach with respect to the lazy Susan), we can substitute the values to find the final angular velocity:
ω = 2π / (2π(0.10 m) / 0.01 m/s) = 1 rad/s

Step 3: Find the speed of the cockroach with respect to the room.
Now that we have the final angular velocity of the lazy Susan, we can find the velocity of the cockroach with respect to the room by adding the linear velocities due to the rotation of the lazy Susan and the velocity of the cockroach with respect to the lazy Susan.

The velocity of the lazy Susan (v_l) is given by:
v_l = Rω,
where R is the radius of the cylinder.

Plugging in the values:
v_l = (0.10 m)(1 rad/s) = 0.10 m/s

The velocity of the cockroach with respect to the room (v_croom) is given by:
v_croom = v_c + v_l,
where v_c is the velocity of the cockroach with respect to the lazy Susan and v_l is the velocity of the lazy Susan.

Plugging in the values:
v_croom = (0.01 m/s) + (0.10 m/s) = 0.11 m/s

Therefore, the speed of the cockroach with respect to the room is 0.11 m/s.

To find the speed of the cockroach with respect to the room, we can use the principle of conservation of angular momentum.

The angular momentum of the system (cockroach + lazy Susan) is conserved since no external torque acts on it. Initially, the angular momentum is zero since both are at rest. When the cockroach starts walking, it begins to rotate the lazy Susan due to its tangential velocity.

The angular momentum of the lazy Susan can be calculated using the equation:
L1 = I * ω1,
where L1 is the angular momentum, I is the moment of inertia of the lazy Susan, and ω1 is the initial angular velocity.

The moment of inertia of the lazy Susan can be calculated using the formula:
I = (1/2) * M * R²,
where M is the mass of the lazy Susan and R is its radius.

Plugging in the given values:
I = (1/2) * 0.24 kg * (0.10 m)²,
I = 0.0024 kg·m².

Since the lazy Susan is initially at rest, the initial angular velocity ω1 is zero, so the initial angular momentum L1 is also zero.

After the cockroach starts walking, it begins to create angular momentum. Since the cockroach is a point mass, its angular momentum is given by:
L2 = m * r * v,
where m is the mass of the cockroach, r is the distance of the cockroach from the axis (10 cm = 0.10 m), and v is its speed with respect to the lazy Susan.

Plugging in the given values:
L2 = 0.015 kg * 0.10 m * 0.01 m/s,
L2 = 0.000015 kg·m²/s.

Since angular momentum is conserved, the sum of the initial and final angular momentum is equal:
L1 + L2 = 0.
0 + 0.000015 kg·m²/s = 0.0024 kg·m² * ω2,
where ω2 is the final angular velocity.

Solving for ω2:
ω2 = (0 + 0.000015 kg·m²/s) / (0.0024 kg·m²),
ω2 = 0.00625 rad/s.

Finally, to find the speed of the cockroach with respect to the room, we can use the formula:
v_room = ω2 * r,
where v_room is the speed of the cockroach with respect to the room and r is its distance from the axis (10 cm = 0.10 m).

Plugging in the values:
v_room = 0.00625 rad/s * 0.10 m,
v_room = 0.000625 m/s.

Therefore, the speed of the cockroach with respect to the room is approximately 0.000625 m/s.

Assume the total angular momentum remains zero (due to frictionless bearing). If the angular velocity of the lazy susan is w, its angular momentum is I*w, where I = (1/2)M*R^2

The angular momentum of the kokroach is equal and opposite to that of the lazy susan.

I*w = m*R*v
where v is the speed of the kokroach with respect to the room, m is the roach mass, and R = 0.10 m.
You also need to use the equation
v - R*w = 0.01 m/s
for the relative velocity

Solve for v.

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