6 Problems on Rotational Dynamics

[chavic]

Homework Statement

The four particles shown below are connected by rigid rods of negligible mass where y = 5.70 m. The origin is at the center of the rectangle.

(a) If the system rotates in the xy plane about the z axis with an angular speed of 6.30 rad/s, calculate the moment of inertia of the system about the z axis.
(b) Calculate the rotational energy of the system.1. Relevant equations
I=$$\sum$$Mi$$r^{2}$$i

And the answer I got is 133.3442

1. The attempt at a solution

And the answer I got for A is 133.3442
But how do I find B?

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2. Homework Statement
Two masses M and m are connected by a rigid rod of length L and of negligible mass, as shown below. For an axis perpendicular to the rod, show that the system has the minimum moment of inertia when the axis passes through the center of mass. Show that this moment of inertia is I = µL2, where µ = mM/(m + M).

Homework Equations

I=$$\sum$$Mi$$r^{2}$$i2. The attempt at a solution
$$x^{2}$$M+$$(L-x)^{2}$$m=mM$$L^{2}$$$$/$$(m+M) ?
AM I on the Right Track?I GOT # 3

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4. Homework Statement
A block of mass m1 = 2.45 kg and a block of mass m2 = 5.55 kg are connected by a massless string over a pulley in the shape of a solid disk having radius R = 0.250 m and mass M = 10.0 kg. These blocks are allowed to move on a fixed block-wedge of angle θ = 30.0°. The coefficient of kinetic friction is 0.360 for both blocks. Draw free-body diagrams of both blocks and of the pulley.

(a) Determine the acceleration of the two blocks.
(b) Determine the tensions in the string on both sides of the pulley.
Left:
Right:

4. Relevant equations
? If someone could just point me in the right direction on the last three...4. The attempt at a solution

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5. Homework Statement
Two blocks, as shown below, are connected by a string of negligible mass passing over a pulley of radius 0.320 m and moment of inertia I. The block on the frictionless incline is moving up with a constant acceleration of 1.20 m/s2.

(a) Determine the tensions in the two parts of the string.
T1:
T2:
(b) Find the moment of inertia of the pulley.

5. Relevant equations
? 5. The attempt at a solution

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6. Homework Statement
The reel shown below has radius R and moment of inertia I. One end of the block of mass m is connected to a spring of force constant k, and the other end is fastened to a cord wrapped around the reel. The reel axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its unstretched position and is then released from rest.

a) Find the angular speed of the reel when the spring is again unstretched. (Answer using theta for θ, g for the acceleration due to gravity, and R, I, m, k, and d, as necessary.)

(b) Evaluate the angular speed numerically at this point if I = 1.20 kg·m2, R = 0.300 m, k = 50.0 N/m, m = 0.500 kg, d = 0.200 m, and θ = 37.0°.

6. Relevant equations

?

6. The attempt at a solution

 

[hage567]

For the first question, just use the formula for rotational energy. $$K_R = \frac{1}{2} I \omega^2$$

 

[hage567]

2. Homework Statement [/b]
Two masses M and m are connected by a rigid rod of length L and of negligible mass, as shown below. For an axis perpendicular to the rod, show that the system has the minimum moment of inertia when the axis passes through the center of mass. Show that this moment of inertia is I = µL2, where µ = mM/(m + M).

Homework Equations

I=$$\sum$$Mi$$r^{2}$$i


2. The attempt at a solution
$$x^{2}$$M+$$(L-x)^{2}$$m=mM$$L^{2}$$$$/$$(m+M) ?
AM I on the Right Track?


[/b]

I don't think you're doing this correctly. What you need to do first is find the centre of mass of the system (i.e. find x in terms of M, m and L). Then use the equation for moment of inertia of two masses (what you have on the left hand side of your equation under "attempt at a solution"). After you've done that, it's just a math exercise. You are trying to show this equation you've set up can be reduced to the one stated in the problem.

? If someone could just point me in the right direction on the last three...

Well, take #4 for example: you need to sum up the torques on the pulley, using $$\Sigma \tau = I \alpha$$. You also need to sum forces on each block in each direction using Newton's second law. Draw your free body diagrams very carefully here. Once you have all of those expressions, you will then be able to combine the information to get the acceleration.