the resultant of two forces is p and q is R.if q is doubled and R is also doubled,if q is reversed.show that p:q:r=2:3:2
p+q = r
p+(-q) = 2r
so,
2p = 3r
see what you can do with that. Not quite sure what you means when you say q is doubled if it is reversed ...
To prove that the ratio of p, q, and r is 2:3:2, we need to consider the effect of each scenario on the resultant.
Let's start by breaking down the initial statement:
- Given: Resultant of two forces = p
- Given: q = R
Scenario 1: q is doubled and R is doubled
In this case, if we double the magnitude of q and R, the resultant becomes:
New resultant = 2q + 2R
We can rewrite this using the given values:
New resultant = 2(q + R) = 2p
Therefore, the ratio of the new resultant to the forces can be expressed as 2:2:2.
Scenario 2: q is reversed
When q is reversed, it means its direction is changed. The new resultant is the difference between the two forces:
New resultant = p - (-q)
Simplifying this expression:
New resultant = p + q
Therefore, the ratio of the new resultant to the forces can be expressed as 1:2:1.
Now let's compare the ratios from both scenarios to the initial ratio:
- Initial ratio: p:q:r = 1:1:1
- Scenario 1 ratio: 2:2:2
- Scenario 2 ratio: 1:2:1
We can see that neither scenario matches the initial ratio. However, to achieve the desired ratio of 2:3:2, we can combine both scenarios by multiplying them together.
Combine Scenario 1 and Scenario 2:
(2:2:2) x (1:2:1) = 2:4:2
Now, if we simplify the combined ratio, we get the desired ratio:
2:4:2 = 2/2 : 4/2 : 2/2
= 1:2:1
Therefore, by combining both scenarios, we have successfully shown that the ratio of p, q, and r is 2:3:2.