Figure shows a rigid rod of length 1.0 m. It is pivoted at O. For what value m, the rod will be in equilibrium ? Find the force (F) exerted on the rod by the pivot. Neglect the weight of the rod.

Answer

To solve the problem step by step, we need to analyze the forces acting on the rod and apply the principles of equilibrium.

### Step 1: Understand the Setup
We have a rigid rod of length 1.0 m pivoted at point O. The rod has two masses attached at points A and B. The distance from O to A is 40 cm (0.4 m) and from O to B is 60 cm (0.6 m). The mass at A is 2 kg, and the mass at B is m kg.

### Step 2: Write the Equilibrium Condition
For the rod to be in equilibrium, the sum of the moments about the pivot point O must be zero. This can be expressed as:

\[ \text{Moment due to } F_1 = \text{Moment due to } F_2 \]

Where:
- \( F_1 = m \cdot g \) (force due to the mass at A)
- \( F_2 = 2 \cdot g \) (force due to the mass at B)

### Step 3: Calculate the Moments
The moment due to \( F_1 \) (at A) about point O is:

\[ \text{Moment}_1 = F_1 \cdot d_1 = (m \cdot g) \cdot 0.4 \]

The moment due to \( F_2 \) (at B) about point O is:

\[ \text{Moment}_2 = F_2 \cdot d_2 = (2 \cdot g) \cdot 0.6 \]

### Step 4: Set the Moments Equal
Setting the two moments equal gives us:

\[ (m \cdot g) \cdot 0.4 = (2 \cdot g) \cdot 0.6 \]

### Step 5: Simplify the Equation
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):

\[ m \cdot 0.4 = 2 \cdot 0.6 \]

Now, calculate the right side:

\[ 2 \cdot 0.6 = 1.2 \]

So, we have:

\[ m \cdot 0.4 = 1.2 \]

### Step 6: Solve for m
Now, solve for \( m \):

\[ m = \frac{1.2}{0.4} = 3 \text{ kg} \]

### Step 7: Find the Force (F) Exerted by the Pivot
To find the force \( F \) exerted on the rod by the pivot, we need to consider the vertical forces acting on the rod. The total downward force is from the mass at B:

\[ F = F_1 + F_2 = (m \cdot g) + (2 \cdot g) = (3 \cdot g) + (2 \cdot g) = 5g \]

Assuming \( g = 10 \, \text{m/s}^2 \):

\[ F = 5 \cdot 10 = 50 \, \text{N} \]

### Final Answers
- The value of \( m \) for equilibrium is **3 kg**.
- The force \( F \) exerted on the rod by the pivot is **50 N**.