Two movers push horizontally on a refrigerator. One pushes due north with a force of 150 N and the other pushes due east with a force of 200 N. (a) Find the direction and magnitude of the resultant force on the refrigerator. (b)Show that the result of the previous problem is independent of the order in which the forces are added.
|F| = sqrt (150^2 + 200^2)
tan theta = 150/200 (angle north of east)
To find the direction and magnitude of the resultant force on the refrigerator, we need to use vector addition.
(a) To calculate the resultant force, we can use the Pythagorean theorem and trigonometry. First, let's visualize the situation:
- The force due north has a magnitude of 150 N and is directed upwards on the y-axis.
- The force due east has a magnitude of 200 N and is directed to the right on the x-axis.
Now, draw a right-angled triangle with the vertical side representing the northern force and the horizontal side representing the eastern force. The resultant force will be the hypotenuse of this triangle.
Using the Pythagorean theorem, we can find the magnitude of the resultant force, R:
R^2 = (150 N)^2 + (200 N)^2
R^2 = 22500 N^2 + 40000 N^2
R^2 = 62500 N^2
Taking the square root of both sides, we find:
R = √(62500) N
R ≈ 250 N
So, the magnitude of the resultant force is approximately 250 N.
To determine the direction of the resultant force, we can use trigonometry. In this case, we can use the tangent function:
tan(θ) = (opposite / adjacent)
θ = tan^(-1)(opposite / adjacent)
θ = tan^(-1)(150 N / 200 N)
Using a calculator or reference table, we can find that θ ≈ 36.87 degrees.
Therefore, the direction of the resultant force is approximately 36.87 degrees north of east.
(b) To show that the result is independent of the order in which the forces are added, we can repeat the calculations with the forces added in reverse order:
- The force due east with a magnitude of 200 N is directed to the right on the x-axis.
- The force due north with a magnitude of 150 N is directed upwards on the y-axis.
Following the same process as before, we find that the magnitude of the resultant force is approximately 250 N, and the direction is approximately 36.87 degrees north of east.
Since the magnitude and direction of the resultant force are the same regardless of the order in which the forces are added, we can conclude that the result is independent of the order.
(a) To find the direction and magnitude of the resultant force on the refrigerator, we can use the concept of vector addition.
Let's consider the force due north as F₁, and the force due east as F₂.
Using the Pythagorean theorem, we can find the magnitude of the resultant force (R) as follows:
R = √(F₁² + F₂²)
= √(150² + 200²)
= √(22500 + 40000)
= √62500
≈ 250 N
To find the direction of the resultant force, we can use the concept of vector addition. The direction of the resultant force can be found by calculating the angle θ using the following equation:
θ = tan⁻¹(F₂ / F₁)
θ = tan⁻¹(200 / 150)
θ ≈ 53.13 degrees
Therefore, the resultant force on the refrigerator has a magnitude of approximately 250 N and a direction of approximately 53.13 degrees with respect to the north.
(b) To show that the result is independent of the order in which the forces are added, let's consider swapping the two forces.
Case 1: F₁ = 150 N (north), and F₂ = 200 N (east) as given in the problem.
Case 2: Let's swap the two forces, so F₁ = 200 N (east), and F₂ = 150 N (north).
Using the same calculations as in part (a), we can find the resultant force for each case.
In Case 1:
R = √(150² + 200²)
≈ 250 N
In Case 2:
R = √(200² + 150²)
≈ 250 N
As we can see, the magnitude of the resultant force is the same in both cases, approximately 250 N. Hence, the result is independent of the order in which the forces are added.