can you tell me what i did wrong on this problem
Rigid rods of negligible mass lying along the y axis connect three particles. The system rotates about the x axis with an angular speed of 3.30 rad/s.
4kg,y=3.00m
2kg,y=-2.00m
3kg,y=-4.00m
a)Find the total rotational kinetic energy evaluated from 1/2 I omega^2
4(3^2)+2(2^2)+3(4^2)=92
.5(92)(3.3)^2
b)Find the tangential speed of each 4kg and 3kg particles
4kg: 4x3.3=13.2m/s
3kg: 3x3.3=9.9m/s
c)Find the total kinetic energy evaluated from Σ mi vi2/2
.5(4kg)(6^2)+.5(2kg)(4^2)+.5(3kg)(8^2)=184
i don't know why i got these wrong
To determine what went wrong, let's go through each part of the problem one by one:
a) Finding the total rotational kinetic energy evaluated from 1/2 I omega^2.
You correctly calculated the moments of inertia for each particle (I = m*r^2) as follows:
4kg particle: I = 4 * (3^2) = 36 kg*m^2
2kg particle: I = 2 * (2^2) = 8 kg*m^2
3kg particle: I = 3 * (4^2) = 48 kg*m^2
However, you made a mistake when summing up the individual rotational kinetic energies. The correct equation is:
Total rotational kinetic energy = (1/2) * Σ I * ω^2
So, the correct calculation would be:
Total rotational kinetic energy = (1/2) * (36 * (3.3^2) + 8 * (3.3^2) + 48 * (3.3^2))
= (1/2) * (36 * 10.89 + 8 * 10.89 + 48 * 10.89)
= (1/2) * (392.04 + 87.12 + 521.28)
= (1/2) * 1000.44
= 500.22 J (rounded to two decimal places)
b) Finding the tangential speed of each 4kg and 3kg particles.
Your calculations for the tangential speeds of the 4kg and 3kg particles are correct:
For the 4kg particle: v = r * ω = 4 * 3.3 = 13.2 m/s
For the 3kg particle: v = r * ω = 8 * 3.3 = 26.4 m/s
c) Finding the total kinetic energy evaluated from Σ mi vi^2/2.
You made a mistake in your calculation for the total kinetic energy using the formula Σ mi * vi^2/2. The correct equation is:
Total kinetic energy = Σ (mi * vi^2)/2
So, the correct calculation would be:
Total kinetic energy = (.5 * (4 * (6^2)) + .5 * (2 * (4^2)) + .5 * (3 * (8^2)))
= (.5 * (4 * 36) + .5 * (2 * 16) + .5 * (3 * 64))
= (.5 * 144 + .5 * 32 + .5 * 192)
= (72 + 16 + 96)
= 184 J
Therefore, your calculation in part (c) is correct.
In conclusion, the mistake you made was in part (a) when summing up the rotational kinetic energies. The correct value for the total rotational kinetic energy is 500.22 J.
To determine what might have gone wrong in your calculations, let's go through each part step-by-step:
a) Total rotational kinetic energy:
You correctly used the formula 1/2 I omega^2, where I represents the moment of inertia and omega is the angular speed.
The moment of inertia for a point mass rotating about an axis is given by I = m r^2, where m is the mass and r is the perpendicular distance from the axis of rotation.
For the 4kg particle, the moment of inertia would be 4 * (3.00^2) = 36.
For the 2kg particle, the moment of inertia would be 2 * (-2.00^2) = 8.
For the 3kg particle, the moment of inertia would be 3 * (-4.00^2) = 48.
Therefore, the total rotational kinetic energy should be:
1/2 * (36 * 3.3^2 + 8 * 3.3^2 + 48 * 3.3^2) = 774.924 (rounded to three decimal places).
b) Tangential speed of each particle:
To find the tangential speed, you correctly multiplied the angular speed by the perpendicular distance from the axis of rotation.
For the 4kg particle, the tangential speed should be 3.3 * 3.00 = 9.9 m/s (not 13.2 m/s as stated).
For the 3kg particle, the tangential speed should be 3.3 * 4.00 = 13.2 m/s (not 9.9 m/s as stated).
c) Total kinetic energy:
You correctly used the formula Σ mi vi^2/2, where mi is the mass of each particle and vi is its tangential speed.
For the 4kg particle, the kinetic energy would be (1/2) * (4 * (9.9^2)) = 193.8 (rounded to one decimal place).
For the 2kg particle, the kinetic energy would be (1/2) * (2 * (0^2)) = 0, as the particle has no tangential speed.
For the 3kg particle, the kinetic energy would be (1/2) * (3 * (13.2^2)) = 275.4 (rounded to one decimal place).
Therefore, the total kinetic energy should be 193.8 + 0 + 275.4 = 469.2 (rounded to one decimal place).
The possible mistakes you made:
1. Made an error in calculating the moment of inertia for each particle.
2. Made an error in calculating the tangential speed for each particle.
3. Made an error in calculating the kinetic energy for each particle.
By reviewing your calculations and applying the correct formulas, you should be able to identify and rectify any errors.