Mechanics particle and tension help please?

Question

A particle P of weight 20 N is suspended by two strings AP and BP. The tension in string AP is (ai + 8j ) and the tension in string BP is (6i+bj).

The value of a is _ and the value of b is

The magnitude of the tension in string AP is

The angle that string BP makes with the vertical is _

Give your answers as exact whole numbers or as decimals to 3 s.f.

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Answer 1 · 2018-03-30T08:42:32

See below.

Calling

#F_1 = (a,8)#
#F_2=(6,b)#
and
#P = (0,-m g)#

we have after equilibrium

#F_1+F_2+P = 0# or

#{(a+6=0),(8+b-m g=0):}#

solving for #a,b# gives

#a = -6#
#b = m g - 8#

now

#T_(AP) = sqrt(a^2+8^2) = sqrt(6^2+8^2)#
#T_(BP) = sqrt(6^2+b^2)= sqrt(6^2+(mg-8)^2)#

the anfle between #F_1, F_2# is obtaining by knowing

#<< F_1, F_2 >> = norm(F_1) norm(F_2) cos phi# then

#phi = arccos((<< F_1, F_2 >> )/(norm(F_1) norm(F_2))) = arccos((a xx 6 + 8 xx b)/(sqrt(a^2+8^2)sqrt(6^2+b^2)))#