To find the minimum height where the force F can be applied so that the crate begins to tip before sliding, we need to consider the factors involved.
When the crate is on the verge of tipping, the static friction force on the bottom of the crate will be at its maximum. The maximum static friction force can be calculated using the equation:
friction force (F_f) = coefficient of static friction (μ) * normal force (F_n)
The normal force (F_n) acting on the crate is equal to the weight of the crate, which can be calculated by multiplying the mass of the crate by the acceleration due to gravity (9.8 m/s²):
F_n = mass (m) * acceleration due to gravity (g)
In this case, the mass of the crate is 16.2 kg, so:
F_n = 16.2 kg * 9.8 m/s²
Next, we need to calculate the maximum static friction force:
F_f = μ * F_n
The coefficient of static friction is given as 0.571, so:
F_f = 0.571 * (16.2 kg * 9.8 m/s²)
Now, we can calculate the torque (Ï„) acting on the crate due to the force F applied at a certain height. For the crate to begin tipping, the torque has to exceed the torque required to keep the crate from sliding.
The torque is given by the equation:
Ï„ = F * h
Where F is the force and h is the height at which the force is applied.
The torque required to keep the crate from sliding is equal to the product of the maximum static friction force and the length of one side of the cube:
Ï„_required = F_f * length of one side (L)
The length of one side of the cube is given as 1.21 m.
So, we have:
τ_required = (0.571 * (16.2 kg * 9.8 m/s²)) * 1.21 m
Since we know that the horizontal force applied to the top of the crate that initiates the tipping is 79.5 N, we can set the torque equal to the torque required to keep the crate from sliding:
F * h = (0.571 * (16.2 kg * 9.8 m/s²)) * 1.21 m
Solving for h, the minimum height at which the force F can be applied:
h = [(0.571 * (16.2 kg * 9.8 m/s²)) * 1.21 m] / 79.5 N
Calculating this expression will give you the minimum height where the force F can be applied so that the crate begins to tip before sliding.