A 15 ft plank of mass 50 kg is laid across two ladders. The ladders are 5 ft apart, and the plank is laid such that 5 ft of it hang off at each end. You have a mass of 90 kg.
(a) How far out past the ladder can you walk before it gets dangerous?
I tried doing
m1r1=m2r2
(33.33)(5)=(90+16.6)r2
and sloved for r2= 1.56, but the computer keeps saying its wrong.
To solve this problem, we can use the concept of torque. Torque is the rotational equivalent of force and is given by the product of the force and the distance from the point of rotation.
Let's start by calculating the torque exerted by the 50 kg plank about the point where it rests on the ladder. The weight of the plank can be considered as acting at its center of mass, which is located 7.5 ft from each end of the plank.
Torque exerted by the plank's weight = (weight of the plank) * (distance to the point of rotation)
= (mass of the plank) * (acceleration due to gravity) * (distance to the point of rotation)
= (50 kg) * (9.8 m/s^2) * (7.5 ft) / (3.28 ft/m)
= 360.98 Nm
As for your mass of 90 kg, it doesn't directly impact the torque acting on the plank. So, we don't need to include it in the calculations.
To keep the system in equilibrium and prevent the plank from tipping over, the torque exerted by your weight must be equal to the torque exerted by the plank's weight. However, since you are walking out from the ladder, your distance from the point of rotation increases, affecting the torque you exert.
Now let's calculate how far out you can walk before it gets dangerous. Let's denote this distance as "x ft." You will be at a distance of (5 + x) ft from the point where the plank rests on the ladder.
Torque exerted by your weight = (your weight) * (acceleration due to gravity) * (distance from the point of rotation)
= (mass of you) * (9.8 m/s^2) * ((5 + x) ft) / (3.28 ft/m)
= (90 kg) * (9.8 m/s^2) * ((5 + x) ft) / (3.28 ft/m)
= 265.85 Nm + (90 kg) * (9.8 m/s^2) * (x ft) / (3.28 ft/m)
Now, to find the maximum distance you can walk before it gets dangerous, you need to set the torque exerted by your weight equal to the torque exerted by the plank's weight:
360.98 Nm = 265.85 Nm + (90 kg) * (9.8 m/s^2) * (x ft) / (3.28 ft/m)
Simplifying the equation, we have:
95.13 Nm = (90 kg) * (9.8 m/s^2) * (x ft) / (3.28 ft/m)
Solve for x:
x = 95.13 Nm * (3.28 ft/m) / ((90 kg) * (9.8 m/s^2))
x ≈ 1.26 ft
Therefore, you can walk out approximately 1.26 ft past the ladder before it gets dangerous.