Definition 1: The linear momentum, vector p, of an object is its mass, m, times its velocity, vector v.
Definition 2: The angular momentum, vector L, of an object about an axis, is defined as the cross product of the position, vector R, relative to the axis, and its linear momentum, vector p.
Definition 3: The magnitude of the cross product is the product of the magnitudes of the vectors times the sine of the angle between their positive directions.
The cross product is a vector perpendicular to both vectors.

Problem:
A marble of mass m moves with velocity v in the x,y plane at angle B relative to position vector R from origin to the mass.
SEE ATTACHMENT #1 PART a for a diagram of the parameters named above.

PART a. From definitions 1 and 2 above, write expressions for the linear momentum and for the angular momentum of the marble about the z axis.

PART b. Write a non-vector equation expressing the magnitude of the marble in terms of its mass, speed, position, and the angle B, then use values as follows to calculate its magnitude: m= .012 kg, R= 1.5 m, v= 48 m/sec, and B= 37 degrees.

PART c. Now the same marble is moving in a circle of radius R about the z axis with the same speed along the circumference. SEE ATTACHMENT #1 PART c. Find the angular momentum about the z axis at the center of the circle.

PART d. Write in terms of the angular velocity, the angular momentum of a mass m moving in a circle of radius R.

Suppose a 60kg person stands at the edge of a 6.0m diameter circular platform, which is mounted on frictionless bearings and has a moment of inertia of 1800kgm^2. The platform and runner are initially at rest. Calculate theangular velocity of the platform if the runner begins to run 4.2m/s.

See attachment please.
Need FBD for each case.
The 0.2 kg ball ( ball is sliding not rotating) and the supporting cord are revolving about the vertical axis on the fixed smooth conical surface with an angular velocity of W = 4 radians/sec. The green ball is held in position b = .3 m by the tension T in the yellow cord.
If b

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Find wf, t

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A uniform solid sphere of radius 0.500 m and mass of 15.0kg turns counterclockwise about a vertical axis through its center. Find its vector angularmomentum when its angular speed is 3.00 rad/s.
I think I have worked this problem correctly. (See attachment) Please check my answer and confirm if in correct please point ou

A space station shaped like a giant wheel has a radius of 105 m and a moment of inertia of 5.03 108 kgm2. A crew of 150 are living on the rim, and the station's rotation causes the crew to experience an apparent free-fall acceleration of g (Fig. P10.44). When 100 people move to the center of the station for a union meeting, the

A child playing marbles shoots a marble directly at another marble at rest. The first marble stops, and the second marble continues in a straight line with the same speed that the first marble had initially. What is the ratio of the masses of the two marbles? I need to see each step and formula to solve this question.

A solid, horizontal cylinder of mass 10.4 kg and radius 1.13 m rotates with an angular speed of 6.81 rad/s about a fixed vertical axis through its center. A 0.251 kg piece of putty is dropped vertically onto the cylinder at a point 0.864 m from the center of rotation, and sticks to the cylinder. Determine the final angular speed