Two forces of 50 and 70 N (newtons) act on an object. The angle between the forces is 40°. Find the magnitude of the resultant and the angle that it makes with the smaller force.

Draw the diagram. Establish an x axis along the direction of the 50N force. break the other force in to x, y componsnts. Add the forces as vectors in x,y directions.

Then use your trig to find the resultant. I will be happy to critique your work.

Fr = 50N[0o] + 70N[40o] = Resultant force.

To find the magnitude of the resultant of two forces, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the magnitudes of those sides multiplied by the cosine of the angle between them.

In this case, let's call the magnitude of the smaller force F1 = 50 N and the magnitude of the larger force F2 = 70 N. The angle between the forces is given as 40°.

Let's calculate the magnitude of the resultant force R using the law of cosines:

R² = F1² + F2² - 2 * F1 * F2 * cos(θ)

where θ is the angle between the forces.

R² = 50² + 70² - 2 * 50 * 70 * cos(40°)

R² = 2500 + 4900 - 2 * 50 * 70 * 0.766

R² = 2500 + 4900 - 5354

R² = 2046

Taking the square root of both sides, we get:

R ≈ √2046

R ≈ 45.26 N (rounded to two decimal places)

So, the magnitude of the resultant force is approximately 45.26 N.

Now, let's calculate the angle that the resultant force makes with the smaller force. We can use the law of sines to find this angle. The law of sines states that in a triangle, the ratio of the length of one side to the sine of the opposite angle is the same for all sides and angles.

sin(θ) / R = sin(θ') / F2

where θ' is the angle the resultant force makes with the smaller force.

Rearranging the equation, we get:

sin(θ') = (F2 / R) * sin(θ)

θ' = arcsin((F2 / R) * sin(θ))

θ' = arcsin((70 / 45.26) * sin(40°))

θ' ≈ arcsin(1.547 * 0.6428)

θ' ≈ arcsin(0.9957)

θ' ≈ 81.04° (rounded to two decimal places)

So, the angle that the resultant force makes with the smaller force is approximately 81.04°.

To find the magnitude of the resultant force, you can use the Law of Cosines. The formula for the magnitude of the resultant force (R) is given by:

R^2 = (F1)^2 + (F2)^2 - 2(F1)(F2)cosθ

Where F1 and F2 are the magnitudes of the forces, and θ is the angle between them. In this case, F1 = 50 N, F2 = 70 N, and θ = 40°.

Plugging in the values into the formula, we get:

R^2 = (50)^2 + (70)^2 - 2(50)(70)cos(40°)

Calculating this equation will give us the magnitude of the resultant force (R).

To find the angle that the resultant force makes with the smaller force, we can use the Law of Sines. The formula for the angle (α) that the resultant force makes with the smaller force is given by:

sin(α) / (F1) = sin(θ) / (R)

Where F1 is the magnitude of the smaller force (50 N), θ is the angle between the forces (40°), and R is the magnitude of the resultant force (which we calculated in the previous step).

Plugging in the values into the formula, we get:

sin(α) / (50) = sin(40°) / (R)

Calculating this equation will give us the angle (α) that the resultant force makes with the smaller force.