An object with a mass of #4 kg# is acted on by two forces. The first is #F_1= < 8 N , -6 N># and the second is #F_2 = < 2 N, 7 N>#. What is the object's rate and direction of acceleration?

The sum of two vectors #< a,b ># and #< c,d ># is #< a+c,b+d>#.

Add the two force vectors #< 8,-6 ># and #< 2,7 ># to get #< 10,1 >#.

The magnitude of a vector #< a,b ># is #sqrt(a^2+b^2)#.

The "size" of the force is #sqrt(10^2+1^2)=sqrt(101)\ "N"#.

According to Newton's second law of motion, the net force acting upon an object is equal to the object's mass times its acceleration, or #F_"net"=ma#. The net force on the object is #sqrt(101)\ "N"#, and its mass is #4\ "kg"#. The acceleration is #(sqrt(101)\ "N")/(4\ "kg")=sqrt(101)/4\ "m"/"s"^2~~2.5\ "m"/"s"^2#.

Newton's first law of motion also states that the direction of acceleration is equal to the direction of its net force. The vector of its net force is #< 10,1 >#.

The angle "theta" of a vector #< a,b ># is #tan(theta)=b/a#.

The angle #theta# of the direction of this vector is #tan(theta)=1/10#. Since both components of the vector are positive, the angle of the vector is in the first quadrant, or #0 < theta < 90^@#. Then, #theta=arctan(1/10)~~5.7^@# (the other possible value, #185.7^@#, is not correct since #0 < theta < 90^@#).

To find the object's acceleration, we first need to calculate the net force acting on it by summing the individual forces. Then, we can use Newton's second law, which states that acceleration is equal to the net force divided by the mass of the object. The net force is obtained by adding the components of the two forces: F_net = F_1 + F_2 = <8 N, -6 N> + <2 N, 7 N> = <8 N + 2 N, -6 N + 7 N> = <10 N, 1 N> Now, we can calculate the magnitude of the net force: |F_net| = sqrt((10 N)^2 + (1 N)^2) = sqrt(100 N^2 + 1 N^2) = sqrt(101 N^2) ≈ 10.05 N Next, we can calculate the direction of the net force: θ = arctan(F_net_y / F_net_x) = arctan(1 N / 10 N) ≈ arctan(0.1) ≈ 5.71 degrees Therefore, the object's rate of acceleration is approximately 10.05 m/s^2 in the direction of approximately 5.71 degrees above the positive x-axis.