An object with a mass of #6 kg# is acted on by two forces. The first is #F_1= < -6 N , 2 N># and the second is #F_2 = < 1 N, 3 N>#. What is the object's rate and direction of acceleration?

#vecF=vecF_1+vecF_2=<-6,2> + <1,3> = <-5,5>#

The mass is #m=6kg#

#vecF=mveca#

#veca=1/mvecF=1/6<-5,5> = <-5/6,5/6>#

#||veca|| =| |<-5/6,5/6>|| = sqrt((-5/6)^2+(5/6)^2)=sqrt(25/36+25/36)=sqrt50/6=1.18ms^-2#

#theta = arctan((5/6)/(-5/6))=arctan(-1)=135^@# antilockwise from the x-axis.

To find the object's rate and direction of acceleration, you first need to calculate the net force acting on the object by summing up the individual forces. Then, you can use Newton's second law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Given: - Mass (\( m \)) = 6 kg - Force \( F_1 = <-6 \, \text{N}, 2 \, \text{N}> \) - Force \( F_2 = <1 \, \text{N}, 3 \, \text{N}> \) First, sum up the forces to find the net force: \[ F_{\text{net}} = F_1 + F_2 = <-6 + 1, 2 + 3> = <-5, 5> \, \text{N} \] Now, calculate the acceleration using Newton's second law: \[ F_{\text{net}} = m \times a \] \[ <-5, 5> = 6 \times a \] Therefore, the acceleration (\( a \)) can be found by dividing the net force by the mass: \[ a = \frac{<-5, 5>}{6} = <-\frac{5}{6}, \frac{5}{6}> \, \text{m/s}^2 \] So, the object's rate of acceleration is \( \frac{5}{6} \) m/s\(^2\) in the positive direction.