The coefficient of static friction between the 2.3 kg crate and the 35.0° incline of Figure P4.47 is 0.241. What minimum force, F, must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline?
ΣF(x) = 0: m•g•sinα=F(fr)
ΣF(y) = 0: m•g•cosα +F = N
F(fr)=μ•N= μ• (m•g•cosα +F)
m•g•sinα= μ• (m•g•cosα +F)
F=(m•g/μ) •(sinα -μ• cosα)
To determine the minimum force required to prevent the crate from sliding down the incline, we need to analyze the forces acting on the crate and use the concept of static friction.
First, let's draw a diagram of the forces acting on the crate.
```
|\
| \
| \
|___\
|__|\
|| || Crate
|| ||
|| ||
-----------------------------------
Incline (35.0°)
```
The weight of the crate acts vertically downward with a force of mg, where m is the mass of the crate (2.3 kg) and g is the acceleration due to gravity (9.8 m/s²).
The force of static friction acts parallel to the incline and opposes the tendency of the crate to slide down. Its magnitude is given by fs = μs * N, where μs is the coefficient of static friction and N is the normal force.
The normal force N is the component of the weight perpendicular to the incline, and it is given by N = mg * cos(θ), where θ is the incline angle (35.0°).
Therefore, the magnitude of the force of static friction is fs = μs * mg * cos(θ).
To prevent the crate from sliding down the incline, the force F applied perpendicular to the incline must be equal to or greater than the force of static friction fs.
Thus, we can write the equation:
F ≥ fs
Substituting the known values:
F ≥ μs * mg * cos(θ)
Now, we can plug in the given values to find the minimum force required to prevent the crate from sliding down the incline.
μs = 0.241 (coefficient of static friction)
m = 2.3 kg (mass of the crate)
g = 9.8 m/s² (acceleration due to gravity)
θ = 35.0° (incline angle)
F ≥ 0.241 * 2.3 kg * 9.8 m/s² * cos(35.0°)
To find the value of cos(35.0°), you can use a scientific calculator or an online calculator.
Once you have the value of cos(35.0°), you can multiply it by the other known values and calculate the minimum force required to prevent the crate from sliding down the incline.