An object with a mass of #9 kg# is acted on by two forces. The first is #F_1= < -2 N , -1 N># and the second is #F_2 = < 8 N, -5 N>#. What is the object's rate and direction of acceleration?

Answer 1

The object is accelerating at a rate of #1.602m/s^2#
at an angle of #-33.69^@#

The total force acting upon an object is the sum of all the forces acting on that object.

#F_"tot"=sumF_i#

In this case

#F_"tot"=F_1+F_2#

Then adding algebraically we get

#F_"tot"= <-2N,1N> + <8N,-5N>#
# = <6N,-4N>#

and by Newton's 2nd Law

#F=ma#
Then we divide by #m=9kg#
#F_"tot"=9kg*<2/3m/s^2,-4/9m/s^2> #

Then

#a=<2/3m/s^2,-4/9m/s^2>#

The rate of acceleration is the magnitude of a.

#|a|=sqrt((2/3)^2+(-4/9)^2)=sqrt(4/9+16/81)=sqrt(36/81+16/81)=sqrt(52/81)=(4sqrt(13))/9approx1.602m/s^2#
The direction is down and to the right and the angle #theta#
#theta=arctan(-2/3)approx.-0.588 radapprox-33.69^@#
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Answer 2
To find the net force, add the individual forces: F_net = F_1 + F_2 = <-2 + 8, -1 - 5> = <6, -6> N Use Newton's second law, F_net = m * a, where m = mass and a = acceleration. For the x-direction: 6 N = 9 kg * a_x, so a_x = 6/9 = 0.67 m/s^2. For the y-direction: -6 N = 9 kg * a_y, so a_y = -6/9 = -0.67 m/s^2. The object's rate of acceleration is 0.67 m/s^2 in the positive x-direction and -0.67 m/s^2 in the negative y-direction.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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