Finding the direction of acceleration in degrees

[kathyt.25]

Homework Statement

"Forces of 10.6 N north, 19.1 N east, and 16.0 N south are simultaneously applied to a 3.62 kg mass as it rests on an air table. What's the direction of acceleration in degrees (Take east to be 0 degrees and counterclockwise to be positive.) "

Using components, I already figured out the magnitude of acceleration.
a=5.48m/s/s
Fnet=19.8N --> broken into components: F(x)=19.1N, F(y)=-5.4N

Homework Equations

Fnet=ma
Trig ratios for sin and cos

The Attempt at a Solution

Using components, I figured out the total force in the x and y directions, and then used the pythagorean theorem to calculate the Fnet.

To find acceleration, I then isolated for a = Fnet/m = 5.48m/s/s

However, I'm having a lot of trouble figuring out the direction of the acceleration, and I don't know why! It's simple trig ratios.

cos(thetha) = F(x)/Fnet = 19.1/19.8
(theta) = 15.3 degrees
Its positive because the counterclockwise direction is positive.

Am I doing it wrong because they are asking for the direction of accleration, and I'm using the direction of Fnet? I don't see how I can use acceleration in the trig ratios though, since acceleration is only in the direction of Fnet... not the components on the x and y axis.

 

[Dick]

You don't need Fnet to find the angle. Use tan(theta)=F(y)/F(x). Using the rounded value of Fnet gives you a fairly large error in this case. The direction of the acceleration is the same as that of the force. So you have the angle relative to the x (east-west) axis. Now look at the force vector. Is that 15 degrees clockwise or counterclockwise from east?

 

[nlightner]

I would like to clarify that the direction of acceleration is not the same as the direction of the net force. Acceleration is a vector quantity that represents the rate of change of an object's velocity, while net force is the vector sum of all forces acting on an object. In this case, the net force is acting in the east direction, but the acceleration may not necessarily be in the same direction.

To find the direction of acceleration, we can use the components of the net force and the mass of the object. In this case, we can use the trigonometric ratios of sine and cosine to calculate the angle of acceleration.

First, we can find the y-component of the net force, which is equal to the negative of the y-component of the force of 10.6 N north. This gives us a y-component of -10.6 N.

Next, we can find the x-component of the net force by subtracting the x-component of the force of 16.0 N south from the x-component of the force of 19.1 N east. This gives us a x-component of 3.1 N.

Now, we can use these components to calculate the angle of acceleration. We can use the tangent ratio, where the opposite side is the y-component and the adjacent side is the x-component. This gives us tan(theta) = (-10.6 N)/(3.1 N), which gives us a theta value of -73.3 degrees. Since we are measuring counterclockwise as positive, we can add 360 degrees to this value to get a final answer of 286.7 degrees. This means that the direction of acceleration is 286.7 degrees counterclockwise from the east direction.

In summary, the direction of acceleration can be found by using the components of the net force and the trigonometric ratios of sine and cosine. It is important to remember that the direction of acceleration may not always be the same as the direction of the net force, as it depends on the mass and the distribution of forces acting on the object.