A balanced beam is pivoted at its center of gravity. A 30 N force is applied at 3 m, and a 40 N force is…

Question

A balanced beam is pivoted at its center of gravity. A 30 N force is applied at 3 m, and a 40 N force is applied at 2 m on one side of the pivot. On the other side, a force X is applied at 3 m. Determ

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The image shows a balanced beam pivoted at its center of gravity. We need to determine the unknown force X and the reaction force R.

To determine X:
We apply the principle of moments, which states that for an object in rotational equilibrium, the sum of clockwise moments about any point is equal to the sum of anticlockwise moments about the same point. We'll take moments about the pivot.

Identify clockwise and anticlockwise moments.
- Anticlockwise moments:
  
  Force of 30 N at 3 m from the pivot: $30 \text{ N} \times 3 \text{ m}$
  
  Force of 40 N at 2 m from the pivot: $40 \text{ N} \times 2 \text{ m}$
- Clockwise moment:
  
  Force of X N at 3 m from the pivot: $X \text{ N} \times 3 \text{ m}$
Set the sum of anticlockwise moments equal to the sum of clockwise moments.
$(30 \text{ N} \times 3 \text{ m}) + (40 \text{ N} \times 2 \text{ m}) = X \text{ N} \times 3 \text{ m}$ $90 \text{ N m} + 80 \text{ N m} = 3X \text{ N m}$ $170 \text{ N m} = 3X \text{ N m}$
Solve for X.
$X = \frac{170 \text{ N m}}{3 \text{ m}}$ $X = 56.67 \text{ N}$
The value of X is $\boxed{56.67 \text{ N}}$ .

To determine R:
For the beam to be in translational equilibrium, the sum of upward forces must equal the sum of downward forces.
Identify upward and downward forces.
- Upward force: R (reaction force at the pivot)
- Downward forces: 30 N, 40 N, and X N
Set the sum of upward forces equal to the sum of downward forces.
$R = 30 \text{ N} + 40 \text{ N} + X \text{ N}$
Substitute the value of X.
$R = 30 \text{ N} + 40 \text{ N} + 56.67 \text{ N}$ $R = 126.67 \text{ N}$
The reaction force R is $\boxed{126.67 \text{ N}}$ .

A balanced beam is pivoted at its center of gravity. A 30 N force is applied at 3 m, and a 40 N force is applied at 2 m on one side of the pivot. On the other side, a force X is applied at 3 m. Determ

Step-by-step solution

Identify clockwise and anticlockwise moments.

Set the sum of anticlockwise moments equal to the sum of clockwise moments.

Solve for X.

Identify upward and downward forces.

Set the sum of upward forces equal to the sum of downward forces.

Substitute the value of X.

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