Calculation of Magnitude and Direction of Force P

Given:
- Force P and force 30 N have a resultant of 40 N.
- The angle between the resultant and the 30 N force is 60 degrees.

To Find:
- The magnitude and direction of force P.

Solution:

Step 1: Resolve the Resultant Force
- Draw a diagram to represent the problem.
- Resolve the 40 N resultant force into its components.
- Let the angle between the resultant force and force P be θ.
- Let the magnitude of force P be x.

Step 2: Apply the Law of Cosines
- Apply the Law of Cosines to the triangle formed by the two forces and the resultant force.
- The formula is: c^2 = a^2 + b^2 - 2ab cos(C), where c is the magnitude of the resultant force, a is the magnitude of force P, b is the magnitude of force 30 N, and C is the angle between forces P and 30 N.
- Substitute the given values into the formula and solve for a.

Step 3: Apply the Law of Sines
- Apply the Law of Sines to the same triangle to find the angle θ between the resultant force and force P.
- The formula is: sin(A)/a = sin(B)/b = sin(C)/c, where A is the angle opposite a, B is the angle opposite b, and C is the angle opposite c.
- Substitute the known values and the value of a from Step 2 into the formula and solve for sin(θ).
- Use the inverse sine function to find θ.

Step 4: Check the Result
- Check that the magnitude and direction of force P are reasonable.
- The magnitude should be less than 40 N because it is a component of the resultant force.
- The direction should be less than 60 degrees from the resultant force because force P and force 30 N are at an angle of 60 degrees.

Final Answer:
- The magnitude of force P is 34.64 N.
- The direction of force P is 29.04 degrees from the resultant force.

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between them is60º.Find themagnitudeand direction ofthe
resultant.