A uniform rod, AB 2 meters long has a mass of 10 kg and rests with its end A on a rough horizontal ground with coefficient of friction 0.6. The rod is is inclined at 30 degrees to the horizontal and is supported on a rough pin at point C, 1.5 meters from A with coefficient of friction 0.3. Draw and calculate all the normal reaction forces.
To calculate the normal reaction forces on the uniform rod, you need to consider the forces acting on it.
First, let's draw a diagram to visualize the situation:
|--- 2m ----|
A--------------B
/|\
/ | \
/ | \
/θ | \
/ | \
/_____|_____\
C 1.5m
Here, A represents the end of the rod resting on the ground. B is the other end, and C is the point where it is supported by a pin.
Since the rod is in equilibrium, the sum of the forces in both the vertical and horizontal directions must be zero.
1. In the vertical direction:
- The weight of the rod acts downward from its center of mass. It can be calculated using the formula: weight = mass * gravity, where gravity ≈ 9.8 m/s².
weight = 10 kg * 9.8 m/s² = 98 N.
- At point A, there is a normal reaction force (R₁) upward, which balances the weight: R₁ = weight = 98 N.
- At point C, there is another normal reaction force (R₂) upward to support the weight of the rod. We need to calculate it.
2. In the horizontal direction:
- At point A, there is a horizontal friction force acting to the left due to the roughness of the ground. We need to calculate this force.
- At point C, there is a horizontal friction force acting to the right due to the roughness of the pin. We also need to calculate this force.
Now let's calculate the unknowns:
1. Friction force at point A:
- The friction force can be calculated using the formula: friction force = coefficient of friction * normal reaction force.
The normal reaction force (R₁) is equal to the weight of the rod, which is 98 N.
So, the friction force at point A = 0.6 * 98 N = 58.8 N.
2. Friction force at point C:
- To calculate the friction force at point C, we need to calculate the horizontal component of the weight:
Horizontal component = weight * sin(30°)
= 98 N * sin(30°)
≈ 49 N.
Now we can calculate the normal reaction force at point C (R₂) by considering the vertical equilibrium:
Vertical forces at point C: R₂ + 49 N - 98 N = 0.
So, R₂ = 49 N + 98 N = 147 N.
Therefore, the normal reaction forces are:
At point A (ground): R₁ = 98 N.
At point C (pin): R₂ = 147 N.
To solve this problem, let's start by drawing a diagram of the situation.
```
|\
C |__\ 1.5 m
| \
| \
| \
| \
| \
| \
---------------------------- \
A B \
```
Now, let's calculate the normal reaction forces at point A and point C.
1. Normal reaction force at point A:
- The weight of the rod is acting vertically downwards, so the normal reaction force at A must balance it out.
- The weight of the rod can be calculated using the equation: weight = mass * gravity.
- In this case, weight = 10 kg * 9.8 m/s^2 = 98 N (Newton).
- Therefore, the normal reaction force at A is 98 N pointing upwards.
2. Normal reaction force at point C:
- The rod is inclined at 30 degrees to the horizontal, so we can break down the weight of the rod into components.
- The component of the weight acting perpendicular to the inclined surface (pointing towards the pin at C) must be balanced by the normal reaction force at C.
- The perpendicular component of the weight can be calculated using the equation: weight * cos(angle of inclination).
- In this case, perpendicular component = 98 N * cos(30 degrees) = 84.58 N (approximately).
- Therefore, the normal reaction force at C is 84.58 N pointing upwards.
To summarize:
- The normal reaction force at point A is 98 N pointing upwards.
- The normal reaction force at point C is 84.58 N pointing upwards.