The mass center g of the slender bar of mass 060 kg and...

The mass center G of the slender bar of mass 0.60 kg and length 0.32 m is falling vertically with a velocity v = 2.1 m/s at the instant depicted. Calculate the angular momentum of the bar about point O if the angular velocity of the bar is (a) $\omega_a$ = 10.5 rad/s clockwise and (b) $\omega_b$ = 10.5 rad/s counterclockwise. The angular momentum is positive if counterclockwise, negative if clockwise. 0 0.50 m- Answers: G 0.32 m $\omega_b$ (a) $H_O$ = i kg-m²/s (b) $H_O$ = i kg-m²/s

Transcript

00:01 We have this system here that rotates with an angular velocity in the counterclockwise direction, and we're asked to find what the angular momentum is about the point g, and also find what the angle is between the angular momentum and the axis from a to b.

00:18 So we're given a weight of the system that's equal to 10 pounds, and we've converted it to mass by doing 10 divided by the acceleration due to gravity, and we'll get a mass that's equal to 0 .3105.

00:37 And these are the equation.

00:38 So to solve this problem, we're going to have to find the different components of the angular momentum.

00:44 So we're going to be using these equations here to help solve for that angular momentum.

00:54 So we'll start the problem by finding what the x component is of the moment of inertia.

01:02 And from the diagram, we can write this as 4 times 1 .1.

01:08 3 times the mass of an individual rod times the length a squared plus 2 times mr times a squared and to get to get this individual mass and r so mr we just take the total mass of the system and divide it by 8 since there's three there's eight different rods on this system so the total mass divided by 8, and this will give us 0 .0 -381.

02:16 And this length a is going to be the length of 10 inches and feet.

02:22 We're going to convert it to feet.

02:25 So i of x is going to be equal to 4 times 1 over 3 times mr, times the length a, which will convert into feet squared plus 2 times mr, times a square.

03:04 So this component is going to be equal to 0 .0 8983.

03:23 And now for the moment of inertia about the xy direction, this is going to be equal to zero.

03:31 Since if we look at the free body diagram, notice that in the xy plane, we don't have any rods, right? like it's all in the xz plane.

03:49 So there's no moment of inertia in this xy plane.

04:00 So now for the xxxie plane, that's going to be equal to two times the mass of a rod times a squared.

04:25 So we just plug in, and now this moment of inertia will be equal to 0 .0 -39 pounds second squared per feet.

04:59 So now we're, so now we can use these equations here.

05:04 Since we're given the angular velocity as 12 radians per second.

05:18 So h of g in the x direction, that's going to be equal to 0 .0 -893 times 12, and that's going to be 1 .07796 pound second feet.

05:50 Now we get our y component, which is going to be 0 times 12, which will just be 0...

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