Calculate Acceleration Magnitude and Direction for Perpendicular Forces
This solution explains how to find the acceleration's magnitude and direction when forces act perpendicularly on an object.
Given Information
- Mass of the body, $m = 5 \text{ kg}$
- First force, $F_1 = 8 \text{ N}$
- Second force, $F_2 = 6 \text{ N}$
- The forces $F_1$ and $F_2$ are mutually perpendicular.
Calculate Net Force
Since the forces are perpendicular, we find the magnitude of the resultant net force ($F_{net}$) using the Pythagorean theorem:
$ F_{net} = \sqrt{F_1^2 + F_2^2} $
Substituting the values:
$ F_{net} = \sqrt{(8 \text{ N})^2 + (6 \text{ N})^2} $
$ F_{net} = \sqrt{64 \text{ N}^2 + 36 \text{ N}^2} $
$ F_{net} = \sqrt{100 \text{ N}^2} $
$ F_{net} = 10 \text{ N} $
Calculate Acceleration Magnitude
Using Newton's second law, $F_{net} = m \times a$, we can find the acceleration ($a$):
$ a = \frac{F_{net}}{m} $
Substituting the values:
$ a = \frac{10 \text{ N}}{5 \text{ kg}} $
$ a = 2 \text{ m s}^{-2} $
Determine Acceleration Direction
The acceleration is in the direction of the net force. We find the angle $\theta$ the net force makes with the $8 \text{ N}$ force.
The tangent of this angle is the ratio of the perpendicular force ($F_2$) to the force it's measured against ($F_1$):
$ \tan(\theta) = \frac{F_2}{F_1} $
$ \tan(\theta) = \frac{6 \text{ N}}{8 \text{ N}} $
$ \tan(\theta) = \frac{3}{4} $
Therefore, the angle is:
$ \theta = \tan^{-1}\left(\frac{3}{4}\right) $
The direction is $\tan^{-1}(3/4)$ with the $8 \text{ N}$ force.
Final Result
The magnitude of the acceleration is $2 \text{ m s}^{-2}$ and its direction is $\tan^{-1}(3/4)$ with the $8 \text{ N}$ force.
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