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Question: The greatest and least resultant of two forces acting at a point is \(10N\) and \(6N\) , respectivel...

The greatest and least resultant of two forces acting at a point is 10N10N and 6N6N , respectively. If each force is increased by 3N3N , find the resultant of new forces when acting at a point at an angle of 90∘90^\circ with each other.

Explanation

Solution

When we have two or more forces, then we can calculate the resultants for the given forces. The greatest resultant is the sum of all the given forces whereas the least resultant is the difference of all the given forces. The net resultant of force will increase or decrease with the increase or decrease in the force's action on it.

Formula Used:
R1=A′2+B′2{R^1} = \sqrt {{A^{'2}} + {B^{'2}}}
Here, R1{R^1}=resultant of the vector and A & BA\,\&\,B= Vectors.

Complete step by step answer:
Let A and B be the two forces. Given that the greatest resultant is 10N10N , for the greatest resultant, we need to add the given two forces A+B=10A + B = 10 .Given that the least resultant is 6N6N , for the least resultant, we need to calculate the difference of the given two forces A−B=6A - B = 6 .
On adding both the resultants we get (A+B)+(A−B)=10+6(A + B) + (A - B) = 10 + 6
On adding the two forces we get 2A=162A = 16
On rearranging it we get A=8A = 8

On substituting the value of the calculated force in the equation of greatest or least resultant equation we can calculate the other force. 8+B=10⇒B=28 + B = 10 \Rightarrow B = 2 or 8−B=6⇒−B=−2⇒B=28 - B = 6 \Rightarrow - B = - 2 \Rightarrow B = 2
Thus the given two forces have the values as A=8A = 8 and B=2B = 2 .
Now, iff force is increased by 3N3N then, A1=8+3=11{A^1} = 8 + 3 = 11 and B1=2+3=5{B^1} = 2 + 3 = 5
As the new forces are acting at an angle of 90∘90^\circ so R1=A′2+B′2{R^1} = \sqrt {{A^{'2}} + {B^{'2}}}
On substituting the values to above equation we get, R1=(11)2+(5)2{R^1} = \sqrt {{{(11)}^2} + {{(5)}^2}}
On squaring the given terms R1=121+25{R^1} = \sqrt {121 + 25}
∴R1=146 N\therefore {R^1} = \sqrt {146} \,N

Hence, the resultant force is 146 N \sqrt {146} \,N.

Note: The net resultant may also be calculated by the principle of superposition (also called principle of imposition), which states that for the calculation of net force (potential or anything else) acting on any object we need to do the sum (addition) of all the forces (potential or anything else) acting on that object.