A 16m long ladder is mounted on a fire truck. The ladder itself has mass 130kg , and at the top is a 32kg basket holding a 83kg firefighter.
If the ladder makes a 60 degrees angle with the horizontal, what's the magnitude of the net torque about the ladder's base?
To find the magnitude of the net torque about the ladder's base, we need to calculate the torque due to the ladder, the basket, and the firefighter separately, and then add them up.
1. Torque due to the ladder:
Since the ladder is mounted on the fire truck, it exerts a force at its center of mass. The torque due to the ladder is given by the formula:
Torque_ladder = force_ladder * distance_ladder
First, let's find the force exerted by the ladder:
The weight of the ladder is given by:
weight_ladder = mass_ladder * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Using the formula for weight, we can find the force:
force_ladder = weight_ladder = mass_ladder * g
Now, let's find the distance from the base of the ladder to its center of mass:
The ladder is 16m long, and it makes a 60-degree angle with the horizontal. The vertical component of the ladder's length is given by:
vertical_component = ladder_length * sin(angle)
horizontal_component = ladder_length * cos(angle)
The distance from the base to the center of mass is half the ladder's length, which is:
distance_ladder = 0.5 * ladder_length
Now we can calculate the torque due to the ladder:
Torque_ladder = force_ladder * distance_ladder
2. Torque due to the basket:
The basket exerts its weight force at the top of the ladder. The torque due to the basket is also given by:
Torque_basket = force_basket * distance_basket
Using the weight formula, we can find the force exerted by the basket:
force_basket = weight_basket = mass_basket * g
Again, let's find the distance from the base of the ladder to the top where the basket is located:
The distance_basket is the length of the ladder.
Now we can calculate the torque due to the basket:
Torque_basket = force_basket * distance_basket
3. Torque due to the firefighter:
Similar to the basket, the torque due to the firefighter is given by:
Torque_firefighter = force_firefighter * distance_firefighter
Using the weight formula, we can find the force exerted by the firefighter:
force_firefighter = weight_firefighter = mass_firefighter * g
And the distance from the base of the ladder to the top is also equal to the ladder's length.
Now we can calculate the torque due to the firefighter:
Torque_firefighter = force_firefighter * distance_firefighter
Finally, we can calculate the magnitude of the net torque by adding up the torques from each component:
Net Torque = Torque_ladder + Torque_basket + Torque_firefighter
Plug in the respective values to calculate the net torque.