An object with a mass of #2 kg# is acted on by two forces. The first is #F_1= < 3 N , 5 N># and the second is #F_2 = < 7 N, -6 N>#. What is the object's rate and direction of acceleration?

The resultant force is

We apply Newton's second Law

The magnitude of the acceleration is

The angle is

To find the object's rate and direction of acceleration, we first need to calculate the net force acting on the object by summing up the individual forces. Then, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. Finally, we can find the acceleration vector by dividing the net force vector by the mass of the object.

Let's denote the mass of the object as ( m = 2 , \text{kg} ). The two forces acting on the object are ( F_1 = \langle 3 , \text{N}, 5 , \text{N}\rangle ) and ( F_2 = \langle 7 , \text{N}, -6 , \text{N}\rangle ).

To find the net force, we sum the individual forces: [ F_{\text{net}} = F_1 + F_2 = \langle 3 , \text{N} + 7 , \text{N}, 5 , \text{N} - 6 , \text{N}\rangle = \langle 10 , \text{N}, -1 , \text{N}\rangle ]

Now, we use Newton's second law to find the acceleration: [ F_{\text{net}} = m \times a ] [ \langle 10 , \text{N}, -1 , \text{N}\rangle = 2 , \text{kg} \times \langle a_x, a_y \rangle ]

Now, we can solve for the acceleration components: [ a_x = \frac{10 , \text{N}}{2 , \text{kg}} = 5 , \text{m/s}^2 ] [ a_y = \frac{-1 , \text{N}}{2 , \text{kg}} = -0.5 , \text{m/s}^2 ]

Therefore, the object's rate of acceleration is (5 , \text{m/s}^2) in the (x)-direction and (-0.5 , \text{m/s}^2) in the (y)-direction.