Only two forces act on an object (mass=4.41 kg), as in the drawing. Find (a) the magnitude and (b) the direction (relative to the x axis) of the acceleration of the object.
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To find the magnitude and direction of the acceleration of the object, we need to analyze the forces acting on it.
Given:
Mass of the object (m) = 4.41 kg
Let's refer to the given forces in the drawing.
1. Force F1: 19.1 N at an angle of 30 degrees above the positive x-axis.
2. Force F2: 10.0 N at an angle of 45 degrees below the negative x-axis.
Step 1: Resolving the forces into x and y components
Let's break down each force into its x and y components using trigonometry.
For Force F1:
Fx1 = F1 * cosθ1
Fy1 = F1 * sinθ1
where θ1 is the angle above the positive x-axis, which in this case is 30 degrees.
For Force F2:
Fx2 = F2 * cosθ2
Fy2 = F2 * sinθ2
where θ2 is the angle below the negative x-axis, which in this case is 45 degrees.
Step 2: Calculate the net force in the x and y directions
Net force in the x-direction (ΣFx) is given by:
ΣFx = Fx1 + Fx2
Net force in the y-direction (ΣFy) is given by:
ΣFy = Fy1 + Fy2
Step 3: Calculate the acceleration
Using Newton's second law (F = ma), we can calculate the acceleration in the x and y directions separately.
Acceleration in the x-direction (ax) is given by:
ax = ΣFx / m
Acceleration in the y-direction (ay) is given by:
ay = ΣFy / m
Step 4: Find the magnitude and direction of the acceleration
The magnitude of acceleration can be found using the Pythagorean theorem:
a = sqrt(ax^2 + ay^2)
The direction of acceleration (θa) can be found using trigonometry:
θa = arctan(ay / ax)
Let's calculate the values:
1. Resolving forces into components:
Fx1 = 19.1 N * cos(30°)
Fy1 = 19.1 N * sin(30°)
Fx2 = 10.0 N * cos(45°)
Fy2 = 10.0 N * sin(45°)
2. Calculating net force:
ΣFx = Fx1 + Fx2
ΣFy = Fy1 + Fy2
3. Calculating acceleration:
ax = ΣFx / m
ay = ΣFy / m
4. Finding magnitude and direction of acceleration:
a = sqrt(ax^2 + ay^2)
θa = arctan(ay / ax)
By following these steps and substituting the values, you can find the magnitude and direction of the acceleration of the object.