"Only two forces act on an object (mass = 3.90 kg), as in the drawing. (F = 76.0 N.) Find the magnitude and direction (relative to the x axis) of the acceleration of the object. " (The angle is 45 degrees and the horizontal force is 40N)
F1 = 76 N @ 45 Deg.
F2 = 40 N. @ 0 Deg.
X = 76*cos45+40*cos(0) = 93.74 N.
Y = 76*sin45 = 53.74 N.
tanA = Y/X = 53.74 / 93.74 = 0.57329.
A = 29.825 Deg.
Fn = X/cosA = 93.74 / cos29.825 = 108.05 N. @ 29.825 Deg.
Fn = ma.
a = Fn/m = 108.05 / 3.90 = 27.71 m/s^2
@ 29.825 Deg.
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To find the magnitude and direction of the acceleration of the object, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
First, let's resolve the given forces into their x and y components:
- The horizontal force is 40 N, and it acts along the x-axis. Therefore, its x-component is 40 N, and its y-component is 0 N.
- The force given in the drawing is 76 N at an angle of 45 degrees relative to the x-axis. To find its x and y components, we need to use trigonometry. The x-component can be calculated as the product of the magnitude of the force and the cosine of the angle: Fx = 76 N * cos(45 degrees). The y-component can be calculated as the product of the magnitude of the force and the sine of the angle: Fy = 76 N * sin(45 degrees).
Next, we can calculate the net force in the x and y directions by adding up the x and y components of the forces:
Net force in the x direction (Fnetx) = Fx + 40 N
Net force in the y direction (Fnety) = Fy + 0 N
Since there are only two forces acting in the x direction, the net force in the x direction is equal to the mass of the object multiplied by its acceleration in the x direction: Fnetx = m * ax.
Similarly, since there are no forces acting in the y direction, the net force in the y direction is equal to zero: Fnety = m * ay = 0.
Now we can set up the equations to solve for the acceleration. We'll use the given mass of 3.90 kg:
Fnetx = m * ax
Fnety = m * ay = 0
Substituting the values we calculated for the net forces and the given mass, we have:
(m * ax) = Fnetx = Fx + 40 N
m * ay = Fnety = 0
Plugging in the known values, we get:
(3.90 kg * ax) = (76 N * cos(45 degrees)) + 40 N
(3.90 kg * ay) = 0
Now we solve these equations to find the acceleration components ax and ay. After finding the values, we can then calculate the magnitude of the acceleration (a) and its direction (relative to the x-axis) using the following formulas:
a = sqrt(ax^2 + ay^2)
θ = atan(ay / ax)
By substituting the calculated values, you will obtain the magnitude and direction (relative to the x-axis) of the acceleration of the object.