Three forces act on a particle that moves with unchanging velocity v = (2 m/s)i – (7 m/s)i. Two of the forces are F1 = (2 N)i + (3 N)j + (– 2N)k and F2 = (– 5N)i + (8 N)j + (– 2 N)k. What is the third force?
The Correct Answer and Explanation is :
To solve for the third force acting on the particle, we need to apply Newton’s second law of motion, which states that the net force acting on an object is equal to the mass of the object times its acceleration: Fnet=m⋅a\mathbf{F_{\text{net}}} = m \cdot \mathbf{a}
However, since the particle is moving with unchanging velocity, its acceleration is zero. This implies that the net force acting on the particle is zero, as there is no change in the velocity of the particle. Therefore: Fnet=0\mathbf{F_{\text{net}}} = 0
Now, the net force is the vector sum of all the forces acting on the particle: F1+F2+F3=0\mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3} = 0
Where:
- F1=(2 N)i^+(3 N)j^+(−2 N)k^\mathbf{F_1} = (2 \, \text{N})\hat{i} + (3 \, \text{N})\hat{j} + (-2 \, \text{N})\hat{k}
- F2=(−5 N)i^+(8 N)j^+(−2 N)k^\mathbf{F_2} = (-5 \, \text{N})\hat{i} + (8 \, \text{N})\hat{j} + (-2 \, \text{N})\hat{k}
- F3\mathbf{F_3} is the unknown force we need to determine.
To find the third force F3\mathbf{F_3}, we rearrange the equation as follows: F3=−(F1+F2)\mathbf{F_3} = – (\mathbf{F_1} + \mathbf{F_2})
Now, we compute the sum of F1\mathbf{F_1} and F2\mathbf{F_2}: F1+F2=[(2+(−5))i^,(3+8)j^,(−2+(−2))k^]\mathbf{F_1} + \mathbf{F_2} = \left[(2 + (-5))\hat{i}, (3 + 8)\hat{j}, (-2 + (-2))\hat{k}\right] F1+F2=(−3 N)i^+(11 N)j^+(−4 N)k^\mathbf{F_1} + \mathbf{F_2} = (-3 \, \text{N})\hat{i} + (11 \, \text{N})\hat{j} + (-4 \, \text{N})\hat{k}
Thus, the third force is the negative of this sum: F3=−(−3i^+11j^−4k^)\mathbf{F_3} = -(-3 \hat{i} + 11 \hat{j} – 4 \hat{k}) F3=(3 N)i^+(−11 N)j^+(4 N)k^\mathbf{F_3} = (3 \, \text{N})\hat{i} + (-11 \, \text{N})\hat{j} + (4 \, \text{N})\hat{k}
Therefore, the third force is: F3=(3 N)i^+(−11 N)j^+(4 N)k^\mathbf{F_3} = (3 \, \text{N})\hat{i} + (-11 \, \text{N})\hat{j} + (4 \, \text{N})\hat{k}
Explanation:
The key concept here is that when an object moves with unchanging velocity, there is no net force acting on it (because acceleration is zero). By applying this principle, we can set the sum of all forces equal to zero. The third force is simply the negative sum of the first two forces, which ensures that the net force remains zero and the velocity remains constant.