A 4.2 kg object is subjected to two forces,
F~
1 = (2.1 N) ˆı + (−1.7 N) ˆ and F~
2 =
(3.2 N) ˆı + (−11.2 N) ˆ. The object is at
rest at the origin at time t = 0.
What is the magnitude of the object’s acceleration?
Fx = 2.1 +3.2 = 5.3
Fy = -1.7 -11.2 = - 12.9
|F| = sqrt ( 5.3^2 + 12.9^2)
|a| = |F|/4.2 meters/second^2
To find the magnitude of the object's acceleration, we need to determine the net force acting on the object and then use Newton's second law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
First, we need to find the net force acting on the object. The net force is the vector sum of the individual forces:
F_net = F_1 + F_2
F_1 = (2.1 N) ˆı + (−1.7 N) ˆ
F_2 = (3.2 N) ˆı + (−11.2 N) ˆ
Let's add these two forces together to find the net force:
F_net = (2.1 N) ˆı + (−1.7 N) ˆ + (3.2 N) ˆı + (−11.2 N) ˆ
Simplifying, we get:
F_net = (2.1 + 3.2) N ˆı + (−1.7 - 11.2) N ˆ
F_net = 5.3 N ˆı + (−12.9) N ˆ
Now that we have the net force, we can calculate the acceleration using Newton's second law. The formula is:
F_net = m * a
Where F_net is the net force, m is the mass of the object, and a is the acceleration.
We are given that the mass of the object is 4.2 kg:
m = 4.2 kg
Substituting the values into the formula:
5.3 N ˆı + (−12.9) N ˆ = 4.2 kg * a
To find the magnitude of the acceleration, we need to find the square root of the sum of the squares of the x and y components of the acceleration:
|a| = sqrt((a_x)^2 + (a_y)^2)
Comparing the x and y components of the equation, we get:
a_x = 5.3 N / 4.2 kg
a_y = -12.9 N / 4.2 kg
Calculating these values:
a_x = 5.3 / 4.2 m/s^2
a_y = -12.9 / 4.2 m/s^2
Now, we can substitute these values into the magnitude equation:
|a| = sqrt((5.3 / 4.2)^2 + (-12.9 / 4.2)^2)
Calculating this expression, we get:
|a| = sqrt(2.35^2 + (-3.07)^2) m/s^2
|a| = sqrt(5.5225 + 9.4249) m/s^2
|a| = sqrt(14.9474) m/s^2
|a| ≈ 3.87 m/s^2
Therefore, the magnitude of the object's acceleration is approximately 3.87 m/s^2.
To find the magnitude of the object's acceleration, we first need to calculate the net force acting on the object and then divide it by the object's mass.
1. Calculate the net force:
The net force is the vector sum of the individual forces acting on the object. To find the net force, add the components of the two forces:
F~net = F~1 + F~2
F~net = [(2.1 N) ˆı + (−1.7 N) ˆ] + [(3.2 N) ˆı + (−11.2 N) ˆ]
F~net = (2.1 N + 3.2 N) ˆı + (−1.7 N - 11.2 N) ˆ
F~net = 5.3 N ˆı - 12.9 N ˆ
2. Divide the net force by the mass:
The magnitude of acceleration (a) is given by Newton's second law of motion: a = Fnet / m, where Fnet is the net force acting on the object and m is the mass of the object.
a = |F~net| / m
a = √[(5.3 N)^2 + (-12.9 N)^2] / 4.2 kg
a = √[28.09 N^2 + 166.41 N^2] / 4.2 kg
a = √194.5 N^2 / 4.2 kg
a = √46.3095 N^2/kg
a ≈ 6.800 m/s^2
Therefore, the magnitude of the object's acceleration is approximately 6.800 m/s^2.