Two forces, one of magnitude 2 N and the other of magnitude 3 N, are applied to the ring of a force table. The directions of both forces are unknown. Which best describes the limitations on R, the resultant? Explain carefully the basis for your answer. (a) R≤5 N (b) 2 N≤R≤3 N (c) R≥3 N (d) 1 N≤R≤5 N (e) R≤2 N.
A
To determine the limitations on the resultant (R), we need to consider the addition of the two forces, each of magnitude 2 N and 3 N, which are applied to the force table.
We know that forces are vectors, meaning they have both magnitude (size) and direction. When adding vectors, the magnitude of the resultant depends on the angle between the vectors. The maximum resultant occurs when the two forces are in the same direction, while the minimum resultant occurs when they are in opposite directions.
Let's consider the extremes:
- If the two forces are in the same direction, their magnitudes add up to give the maximum resultant. In this case, the resultant would be 2 N + 3 N = 5 N.
- If the two forces are in opposite directions, their magnitudes subtract to give the minimum resultant. In this case, the resultant would be |2 N - 3 N| = 1 N.
Therefore, we have determined that the range of the resultant (R) is between 1 N and 5 N.
Looking at the answer choices:
(a) R ≤ 5 N - This is true because the maximum resultant is 5 N.
(b) 2 N ≤ R ≤ 3 N - This is not true since the minimum resultant is 1 N and the maximum resultant is 5 N.
(c) R ≥ 3 N - This is true because the minimum resultant is 1 N.
(d) 1 N ≤ R ≤ 5 N - This is true as it includes both the minimum and maximum resultant values.
(e) R ≤ 2 N - This is not true since the maximum resultant is 5 N.
Given the limitations we derived, the best answer is (d) 1 N ≤ R ≤ 5 N.
To determine the limitations on the resultant, R, of the two unknown forces of magnitude 2 N and 3 N applied to the force table, we can use the principle of vector addition. The resultant is the vector sum of the two forces.
When adding vectors, the magnitudes of the resultant will always be between the sum and difference of the magnitudes of the individual vectors. So, we can add the magnitudes of the forces to determine the minimum and maximum possible magnitudes of the resultant.
The minimum possible magnitude of the resultant occurs when the two forces are acting in the opposite direction, subtracting from each other. In this case, the minimum magnitude of the resultant will be the difference between the magnitudes of the individual forces: Rmin = |2 N - 3 N| = 1 N.
The maximum possible magnitude of the resultant occurs when the two forces are acting in the same direction, adding to each other. In this case, the maximum magnitude of the resultant will be the sum of the magnitudes of the individual forces: Rmax = 2 N + 3 N = 5 N.
Therefore, the limitations on the resultant, R, are 1 N ≤ R ≤ 5 N. So the correct answer is (d) 1 N ≤ R ≤ 5 N.