Two forces are applied to a car in an effort to move it, as shown in the following figure, where F1 = 453 N and F2 = 400 N. (Assume up and to the right as positive directions.)(F1 is 8degrees and F2 is 35degreed)
(a) What is the resultant of these two forces?
magnitude
direction
(b) If the car has a mass of 3805 kg, what acceleration does it have? Ignore friction.
Fx = 453 cos 8 + 400 cos 35
Fy = 453 sin 6 + 400 sin 35
|F| = sqrt ( Fx^2 + Fy^2)
tan theta = Fy/Fx where theta is angle above x axis
|A| = |F| / 3805
same angle above x axis = theta
In order to find the resultant of the two forces applied to the car, we first need to find the components of each force in the x and y directions.
(a) To find the resultant magnitude, we can use the Pythagorean theorem, which states that the magnitude of the resultant of two perpendicular vectors is equal to the square root of the sum of the squares of their magnitudes.
Let's calculate the x and y components of the forces:
For the force F1:
Fx1 = F1 * cos(angle1)
Fx1 = 453 N * cos(8°)
Fx1 ≈ 449.28 N (rounded to two decimal places)
Fy1 = F1 * sin(angle1)
Fy1 = 453 N * sin(8°)
Fy1 ≈ 62.46 N (rounded to two decimal places)
For the force F2:
Fx2 = F2 * cos(angle2)
Fx2 = 400 N * cos(35°)
Fx2 ≈ 326.36 N (rounded to two decimal places)
Fy2 = F2 * sin(angle2)
Fy2 = 400 N * sin(35°)
Fy2 ≈ 228.67 N (rounded to two decimal places)
Now, let's add the x and y components separately:
Resultant in the x-direction: Rx = Fx1 + Fx2
Rx = 449.28 N + 326.36 N
Rx ≈ 775.64 N (rounded to two decimal places)
Resultant in the y-direction: Ry = Fy1 + Fy2
Ry = 62.46 N + 228.67 N
Ry ≈ 291.13 N (rounded to two decimal places)
Now, we can use the Pythagorean theorem to find the resultant magnitude:
Resultant magnitude: R = sqrt(Rx^2 + Ry^2)
R = sqrt((775.64 N)^2 + (291.13 N)^2)
R ≈ 830.97 N (rounded to two decimal places)
To find the direction of the resultant, we can use the inverse tangent function:
Resultant direction: angle = arctan(Ry / Rx)
angle = arctan(291.13 N / 775.64 N)
angle ≈ 20.72° (rounded to two decimal places)
Therefore, the resultant of the two forces has a magnitude of approximately 830.97 N and a direction of approximately 20.72°.
(b) To determine the acceleration of the car, we can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass.
Acceleration (a) = Net force / Mass
The net force acting on the car is the resultant force (R), which has already been calculated: R ≈ 830.97 N.
Given that the mass of the car (m) is 3805 kg:
Acceleration (a) = 830.97 N / 3805 kg
Acceleration (a) ≈ 0.218 m/s² (rounded to three decimal places)
Therefore, the car has an acceleration of approximately 0.218 m/s².