Find the angle of acceleration in circular motion

[trivk96]

Homework Statement

Find the direction of acceleration relative to
the direction of motion. Answer between −180°
and 180°.

The tangential acceleration is .98 m/s². The centripital acceleration is 2.205m/s²


Answer in units of °

Homework Equations

?

The Attempt at a Solution

I tried tan( 2.205/.98) but this does not seem right at all. Could you also explain how you got the answer and why it works.


Thanks in advance

 

[LawrenceC]

Don't you mean arctan rather than tan? Tan need degrees. 2.205/.98 is not degrees.

 

[trivk96]

So does that mean the answer is 66.03°

 

[LawrenceC]

That is the arctan of 2.205/.98 so you computed the correct angle. It's 66.04 degrees measured from the tangential acceleration vector. It is pointing in the general direction of tangential acceleration but more so towards the inside of the arc because the magnitude of the centripital acceleration is the larger of the two accelerations. If you draw the vectors to scale and place them head to tail and close the right triangle, you'll see where the resultant it is pointing.

 

[asteriatic]

The angle of acceleration in circular motion can be found using the equation:

θ = tan-1 (aT/aC)

Where θ is the angle of acceleration, aT is the tangential acceleration, and aC is the centripetal acceleration.

In this case, θ = tan-1 (0.98/2.205) = 24.9°

This means that the direction of acceleration is 24.9° clockwise from the direction of motion. This makes sense because in circular motion, the direction of acceleration is always perpendicular to the direction of motion, and in this case, it is also directed towards the center of the circle.

It is important to note that the angle of acceleration is always measured from the direction of motion, so it can be negative if the acceleration is in the opposite direction. In this case, the angle would be -24.9°.

It is also important to note that the tangential and centripetal accelerations are perpendicular to each other, so the angle between them will always be 90°. This can be seen by using the Pythagorean theorem:

aT^2 + aC^2 = a^2

Where a is the total acceleration. In this case, a = √(0.98^2 + 2.205^2) = 2.38 m/s^2. So the angle between the two accelerations can also be found using the inverse cosine function:

θ = cos-1 (aC/a) = cos-1 (2.205/2.38) = 41.8°

This means that the centripetal acceleration is 41.8° counterclockwise from the tangential acceleration. Again, the negative sign indicates the direction of acceleration relative to the direction of motion.

I hope this explanation helps you understand the concept of finding the angle of acceleration in circular motion. It is important to remember to always use the inverse trigonometric functions when dealing with angles in physics, and to pay attention to the direction of acceleration relative to the direction of motion.