Combining rotational and translational velocities

In summary, we need to use the formula v = w X r to find the translational velocity of the wrist in this problem, which is (1, -4, 2). We can assume that the translational velocity of the elbow is caused by the rotational velocity of the forearm.
  • #1
aplysia
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Hi, I need help with a homework problem, because I am unsure in what way rotational and translational velocity combine.
Also, I can not really imagin ethe movement described in the problem, so I am not sure how to start

Homework Statement


At some instant a person’s forearm has a rotational velocity of (2, 1, 4), the elbow has translational
velocity (1, 0, 0) and the wrist is at location (0, 1, 1) relative to the elbow. What is the translational
velocity of the wrist?

Homework Equations


v = w X r

The Attempt at a Solution


when substituting the rotational velocity for w, the velocity and the velocity of the elbow for v, I get a system of 3 equations with three unknown variables that can not be solved.
So I know that translational velocities just add by vector addition, but that rotational velocities are non-commutative, so I can not just add them. But in this case, one vector is rotational, so I don't know how to combine this two velocity vectors.
Also, I am not sure about the given date, because it is not stated if the translational velocity of the elbow is caused by the rotational velocity of the forearm.
So probably I miss something quite obvious here, but your help would be really appreciated,
thanks in advance
 
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  • #2


Dear student,

Thank you for reaching out for help with your homework problem. It seems like you are struggling with understanding how rotational and translational velocity combine. Let's break down the problem and see if we can come up with a solution together.

First, let's define what rotational and translational velocity mean. Rotational velocity refers to the speed at which an object is rotating around a fixed point or axis. Translational velocity, on the other hand, refers to the speed at which an object is moving in a straight line.

In this problem, we are given the rotational velocity of the forearm (2, 1, 4) and the translational velocity of the elbow (1, 0, 0). We are also given the location of the wrist relative to the elbow (0, 1, 1). To find the translational velocity of the wrist, we need to use the formula v = w X r, where v is the translational velocity, w is the rotational velocity, and r is the location of the object relative to the point of rotation.

In this case, we are looking for the translational velocity of the wrist, so we can substitute the given values into the formula as follows:

v = (2, 1, 4) X (0, 1, 1)

To calculate the cross product, we can use the following formula:

(a, b, c) X (d, e, f) = (bf-ce, cd-af, ae-bd)

Using this formula, we get:

v = (1, -4, 2)

So, the translational velocity of the wrist is (1, -4, 2).

Now, to address your concern about the given data, we can assume that the translational velocity of the elbow is caused by the rotational velocity of the forearm. This is because the forearm is connected to the elbow and the rotation of the forearm would result in the elbow moving in a straight line.

I hope this helps clarify the problem for you. If you have any further questions, please don't hesitate to ask. Good luck with your homework!
 

1. What is the difference between rotational and translational velocities?

Rotational velocity refers to the speed at which an object rotates around an axis, while translational velocity refers to the speed at which an object moves in a straight line.

2. How do rotational and translational velocities combine to determine an object's total velocity?

The total velocity of an object is determined by combining its rotational and translational velocities using vector addition. This means that the two velocities are added together, taking into account their direction and magnitude.

3. How do we measure rotational and translational velocities?

Rotational velocity is typically measured in radians per second (rad/s) or revolutions per minute (rpm), while translational velocity is usually measured in meters per second (m/s) or kilometers per hour (km/h). These measurements can be obtained using various instruments such as speedometers, tachometers, and gyroscopes.

4. What are some real-life examples of objects with combined rotational and translational velocities?

A spinning top is a great example of an object with both rotational and translational velocities. As it spins around its axis, it also moves across a surface, combining both types of velocity. Other examples include a wheel rolling down a hill and a spinning figure skater moving across the ice.

5. How do rotational and translational velocities affect an object's stability?

Rotational velocity can affect an object's stability by causing it to spin or rotate, while translational velocity can affect its stability by causing it to move in a certain direction. The interaction between these two velocities can determine how stable an object is and how well it maintains its balance.

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