Three objects lie in the x, y plane. Each rotates about the z axis with an angular speed of 4.44 rad/s. The mass m of each object and its perpendicular distance r from the z axis are as follows: (1) m1 = 6.00 kg and r1 = 1.00 m, (2) m2 = 4.50 kg and r2 = 7.10 m, (3) m3 = 3.60 kg and r3 = 2.80 m. Find the tangential speed of (a) object 1, (b) object 2, and (c) object 3. (d) Determine the total kinetic energy of this system using the expression (e) Find the rotational kinetic energy of the system using the relation to verify that the answer is the same as that in (d).
To find the tangential speed of each object, we can use the formula:
v = ω * r
Where v is the tangential speed, ω is the angular speed, and r is the perpendicular distance from the z-axis.
(a) For object 1:
v1 = ω * r1
= 4.44 rad/s * 1.00 m
= 4.44 m/s
So, the tangential speed of object 1 is 4.44 m/s.
(b) For object 2:
v2 = ω * r2
= 4.44 rad/s * 7.10 m
= 31.484 m/s
So, the tangential speed of object 2 is 31.484 m/s.
(c) For object 3:
v3 = ω * r3
= 4.44 rad/s * 2.80 m
= 12.432 m/s
So, the tangential speed of object 3 is 12.432 m/s.
Moving on to the next part:
(d) To determine the total kinetic energy of the system, we can use the formula:
Total Kinetic Energy = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 + (1/2) * m3 * v3^2
Substituting the values we have:
Total Kinetic Energy = (1/2) * 6.00 kg * (4.44 m/s)^2 + (1/2) * 4.50 kg * (31.484 m/s)^2 + (1/2) * 3.60 kg * (12.432 m/s)^2
Calculating this expression will give you the total kinetic energy of the system.
(e) Finally, to find the rotational kinetic energy of the system, we can use the formula:
Rotational Kinetic Energy = (1/2) * I * ω^2
Where I is the moment of inertia.
The moment of inertia for each object can be calculated using the formula:
I = m * r^2
Substituting the values we have:
I1 = 6.00 kg * (1.00 m)^2
I2 = 4.50 kg * (7.10 m)^2
I3 = 3.60 kg * (2.80 m)^2
Then, we can calculate the rotational kinetic energy of the system using the formula and compare it with the total kinetic energy calculated in (d).
If the values match, it verifies that the calculation is correct.