Two forces P and Q act such that the resultant R is equal to P. if P is doubled, find the angle made by the new resultant with respect to the force Q.

The resultamt of the forces P and Q is R. If Q is doubled then R is doubled. If Q is reversed, even then R is doubled. Find the ratio P:Q:R.

To tackle this problem, let's break down the information provided step by step, focusing on the relationships between the forces P, Q, and the resultant R. We know that initially, the resultant R is equal to the force P. Now, let's analyze what happens when we manipulate the forces.

Initial Setup

We start with two forces, P and Q, where the resultant R is given by:

• R = P

This implies that the forces are acting in such a manner that the magnitude of the resultant equals the magnitude of P.

Doubling Force P

Now, if we double the force P, we have:

• P' = 2P

The new resultant, denoted R', will depend on the angle θ between forces P and Q. We can use the law of cosines to express R' as follows:

• R' = √(P'^2 + Q^2 + 2 * P' * Q * cos(θ))

Substituting P' into the equation gives:

• R' = √((2P)^2 + Q^2 + 2 * (2P) * Q * cos(θ))

Finding the Angle

Next, we need to determine the angle that this new resultant makes with respect to force Q. When P is doubled, we can analyze the resultant in terms of the new configuration. Notice that if Q is doubled, the resultant changes, which leads us to explore the implications of these changes.

Effect of Doubling Q

If we double Q, the new force becomes:

• Q' = 2Q

In this case, the resultant will be:

• R'' = √(P^2 + (2Q)^2 + 2 * P * (2Q) * cos(θ'))

Reversing Q

When Q is reversed, the force now acts in the opposite direction. The resultant can be calculated similarly, leading to:

• R''' = √(P^2 + Q^2 - 2PQ * cos(θ'))

In both scenarios where Q is doubled or reversed, we see that the resultant's magnitude is affected. This indicates a relationship where the resultant can remain consistent or even double under certain conditions.

Finding the Ratios P:Q:R

Now, let's summarize the relationships we've found:

• If Q is doubled, we can infer that the resultant also doubles.
• If Q is reversed, the resultant is still effectively doubled due to the direction change.

To find the ratio P:Q:R, let’s assume:

• P = k
• Q = m
• R = P = k

From this, we can derive the ratios based on the relationships established:

• R = k
• If Q = k, then R remains tied to P.

Thus, if we establish a proportionality, we can state:

• P : Q : R = k : m : k

In simplified terms, since R equals P, we can express the ratio succinctly as:

• P : Q : R = 1 : 1 : 1

This leads us to conclude that under the various manipulations of forces P and Q, the ratio remains consistent for the resultant R. Understanding these relationships is crucial in vector analysis and mechanics, providing insight into how forces interact in a system.