To solve for the values of $P$ and $Q$ in this equilibrium problem, we need to apply the conditions for equilibrium. The forces are balanced, meaning both the horizontal and vertical components of the forces must sum to zero.

Step 1: Resolve the forces into components

The given forces are:

• A force of 10 N at an angle of 49° above the horizontal.
• Forces $P$ and $Q$, whose directions are given as vertical and angled 18° below the horizontal, respectively.

Force Components:

• 10 N Force:

  • Horizontal component: $10 \cos(49^\circ)$
  • Vertical component: $10 \sin(49^\circ)$

• Force $Q$ (angled at 18° below the horizontal):

  • Horizontal component: $Q \cos(18^\circ)$
  • Vertical component: $Q \sin(18^\circ)$

• Force $P$ is purely vertical, so it only has a vertical component $P$.

Step 2: Set up equilibrium conditions

Horizontal equilibrium:

Since the system is in equilibrium, the sum of horizontal forces must be zero: $10 \cos(49^\circ) = Q \cos(18^\circ)$ From this, we can solve for $Q$: $Q = \frac{10 \cos(49^\circ)}{\cos(18^\circ)}$

Vertical equilibrium:

The sum of the vertical forces must also be zero: $P + Q \sin(18^\circ) = 10 \sin(49^\circ)$ We can rearrange this to solve for $P$: $P = 10 \sin(49^\circ) - Q \sin(18^\circ)$

Step 3: Calculation

Using the known trigonometric values:

• $\cos(49^\circ) \approx 0.6561$
• $\sin(49^\circ) \approx 0.7547$
• $\cos(18^\circ) \approx 0.9511$
• $\sin(18^\circ) \approx 0.3090$

Solving for $Q$:

$Q = \frac{10 \times 0.6561}{0.9511} \approx 6.90 \, \text{N}$

Solving for $P$:

$P = 10 \times 0.7547 - 6.90 \times 0.3090 \approx 7.55 - 2.13 = 5.42 \, \text{N}$

Final Answer:

• $Q = 6.9 \, \text{N}$
• $P = 5.4 \, \text{N}$

Would you like a detailed explanation of the steps or any clarifications?

Here are 5 follow-up questions:

1. What happens if the angle of force $Q$ is changed?
2. How do we solve for forces in non-equilibrium cases?
3. Can this method be used for three-dimensional force problems?
4. How do the trigonometric identities simplify force resolution?
5. What happens if one of the forces is doubled?

Tip: Always break forces into their components when dealing with equilibrium problems involving angled forces.