# General method for approaching block/rotating pulley/rotating object equations

• Same-same
In summary, the general method to solve problems involving tension, rotating objects, rotating pulleys, and gravitational forces is to draw free body diagrams, identify and label forces, write Newton's Second Law equations for each component, establish relations between linear and angular accelerations, and solve for unknowns. Translational velocity should be included in calculations when the center of mass of a rotating object is translating relative to the frame of reference.

#### Same-same

Not asking about a specific problem, but general methods. I'm having trouble with problems that usually involve tension, rotating objects, rotating pulleys, and one block free to fall due to gravity. I'm not asking for help with a specific problem, but with this approach in general. It seems like it should be simple, but for whatever reason I always freeze up whenever I try to solve one of these problems, so I'm trying to better understand rotation.

My current line of reasoning is to use force diagrams and then
Since $\tau$ = RF sin ($\theta$) =I$\alpha$,
F =((I$\alpha$)/(R sin($\theta)$))

Since the net force is generally provided by one or more objects allowed to fall
ƩF$_{y}$ (generally equal to m$_{falling object}$g- F$_{tension}$ )=m$_{total}$a + ((I $_{1}$ $\alpha$)/ (R$_{1}$ sin ($\theta)$))
+ ((I $_{2}$ $\alpha$))/ (R$_{2}$ sin ($\theta$) ))... and so on for any of the other I$\alpha$ and ma.
I will then generally plug in a/R for $\alpha$

At this point, I'm generally not sure what to do. Possible ideas are to do a sum of torques equation, or to attempt to solve for Tension in the x direction.

Another thing that confuses me is when to include translational velocity for a rotating object in my equations.

The general method is to
1. Draw free body diagrams (FBDs) for each identifiable component of the system. For example, if you have two masses hanging on either side of a pulley that is not assumed massless, you will need three separate FBDs, two for the hanging masses and one for the pulley.
2. Identify the forces in each diagram and label them appropriately. This means use different labels for different forces, however if ##F_1## in one FBD has a reaction counterpart in another, then they should be labeled the same and point in opposite directions in the diagram; this is the case most notably with tension. Also note that the tension on either side of a pulley with mass (and hence moment of inertia) has to be labeled differently, otherwise the pulley will not acquire angular acceleration.
3. Based on each of the FBDs write Newton's Second Law ##F_{net,i}=m_i a_i## or ##\tau_{net,i}=I_i\alpha_i## for each of the components of the system.
4. Establish relations between linear accelerations and between linear accelerations and angular accelerations. These will depend on how the total system is assembled.
5. The Newton's Second Law equations should be sufficient to solve for the unknowns.

You include translational velocity for a rotating object when the center of mass of the object is translating relative to your frame of reference. For example, if you are riding a unicycle, you do not include the translational velocity in the calculation of its wheel's kinetic energy. If you are standing by the side of the road and watch the unicycle go by, you do.

## 1. What is a block/rotating pulley/rotating object equation?

A block/rotating pulley/rotating object equation is a mathematical representation of the relationship between forces, masses, and acceleration in a system involving blocks, pulleys, and rotating objects. It is used to predict the motion of these objects and determine the forces acting on them.

## 2. How do you approach solving these types of equations?

The general method for approaching these equations involves identifying all the forces acting on the objects, setting up equations based on Newton's laws of motion, and solving for the unknown variables. It is important to draw a free-body diagram to visualize the forces and use the appropriate equations for each object in the system.

## 3. What are some common assumptions made when using these equations?

Some common assumptions when using these equations include neglecting air resistance, assuming that the pulley is massless and frictionless, and assuming that the string or rope is massless and does not stretch. These assumptions simplify the equations and make them easier to solve.

## 4. Can these equations be used for any type of system involving blocks, pulleys, and rotating objects?

Yes, these equations can be used for any type of system involving these objects, as long as the assumptions hold true. However, more complex systems may require additional equations and considerations, such as rotational inertia and torque.

## 5. How do these equations relate to real-world applications?

These equations have many real-world applications, such as in engineering, physics, and mechanics. They can be used to design and analyze machines and structures involving pulleys, such as cranes and elevators, and to understand the motion of rotating objects, such as wheels and gears.