An object with a mass of #2 kg# is acted on by two forces. The first is #F_1= <4 N , 8 N># and the second is #F_2 = < 1 N, 7 N>#. What is the object's rate and direction of acceleration?

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Answer 1 · 2017-01-04T02:42:44

The objects rate of acceleration is #7.91 \ ms^-2# (2dp)
The direction of the acceleration is at an angle of #71.6º# (1dp)

Let the resultant force acting on the object be # vec F #, Then

# vecF = vecF_1 + vecF_2#
# \ \ \ \ = <<4,8>> + <<1,7>>#
# \ \ \ \ = <<5,15>>#

The magnitude, #F# of # vecF # is given by it's norm;

# F = | vecF | #
# \ \ \ \= |<<5,15>>| #
# \ \ \ \= sqrt(5^2+15^2) #
# \ \ \ \= sqrt(25+225) #
# \ \ \ \= sqrt(250) #
# \ \ \ \= 15.81 \ N # (2dp)

Applying Newton's 2nd Law of Motion, # F=ma#, we have:

# :. 15.81=2a#
# :. a=7.91 \ ms^-2# (2dp)

The direction of the acceleration is the same as the direction of the resultant force #vecF#. If this is at an angle #theta# then

# tan theta = 15/5 #
# :. \ theta = arctan 3#
# :. \ theta = 71.6º# (1dp)

Hence,

The objects rate of acceleration is #7.91 \ ms^-2# (2dp)
The direction of the acceleration is at an angle of #71.6º#