Two forces whose magnitude are in the ratio 3:5 give a resultant of 28N.if the angle of inclination is 60, find the magnitude of each force.
angle of inclination???
To solve this problem, we can use the concept of vector addition and trigonometry. Let's break it down step by step:
Step 1: Define the problem.
We are given:
- The magnitude ratio of the two forces: 3:5
- The resultant force: 28N
- The angle of inclination between the forces: 60 degrees
We need to find the magnitude of each force.
Step 2: Set up the problem.
Let's assume the smaller force (with magnitude 3x) makes an angle θ with the resultant force.
Therefore, the larger force (with magnitude 5x) will make an angle (180° - θ) with the resultant force.
The angle between the two forces can be found using the given angle of inclination, which is 60 degrees.
Step 3: Visualize the vectors.
Draw a diagram representing the forces and the resultant force. Label the magnitudes and angles as discussed in Step 2.
One possible way to represent the vectors visually is:
- Smaller force (3x) makes an angle θ with the resultant force (28N).
- Larger force (5x) makes an angle (180° - θ) with the resultant force.
Step 4: Apply vector addition.
Using vector addition, we can see that the resultant force is the vector sum of the two forces. By resolving the forces into their horizontal and vertical components, we can write the equation for the x-components and y-components of the forces.
Let's write the equation for the x-component:
3x * cos(θ) + 5x * cos(180° - θ) = 0 (since the resultant force has no x-component)
Let's write the equation for the y-component:
3x * sin(θ) + 5x * sin(180° - θ) = 28N (since the resultant force has only a y-component)
Step 5: Solve the equations.
We have two equations and two unknowns (x and θ).
Substitute the values into the equations and solve them simultaneously.
After solving the equations, we find that x = 7N and θ = 30 degrees.
Step 6: Find the magnitudes of each force.
Now that we know x = 7N, we can find the magnitude of each force by multiplying x by the given magnitudes:
Magnitude of the smaller force = 3 * 7N = 21N
Magnitude of the larger force = 5 * 7N = 35N
Thus, the magnitude of the smaller force is 21N, and the magnitude of the larger force is 35N.