The coefficient of static friction between the 3.00 -kg crate and the 35 degree incline is .300. What minimum force must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline.
To calculate the minimum force required to prevent the crate from sliding down the incline, we need to consider the components of forces acting on the crate.
1. Weight force (Fg): The weight of the crate can be calculated as the product of its mass (m) and the acceleration due to gravity (g). The formula is Fg = m * g.
2. Normal force (Fn): The normal force is the perpendicular force exerted by the incline on the crate. It is equal in magnitude but opposite in direction to the component of the weight force perpendicular to the incline. The formula is Fn = Fg * cos(theta), where theta is the angle of the incline.
3. Friction force (Ff): The static friction force will oppose the motion of the crate and prevent it from sliding down the incline. The formula for the static friction force is Ff = mu * Fn, where mu is the coefficient of static friction.
4. Applied force (Fa): The force perpendicular to the incline that we want to find.
Now, we can set up the equation for equilibrium along the direction perpendicular to the incline:
Fa - Ff - Fn = 0
Since the crate is not sliding, the force of friction is at its maximum value, which is equal to the coefficient of static friction multiplied by the normal force. Therefore:
Fa - (mu * Fn) - Fn = 0
Substituting the values we have:
Fa - (mu * Fn) - Fn = 0
Fa - (mu * (Fg * cos(theta))) - (Fg * cos(theta)) = 0
Solving for Fa:
Fa = (mu * (Fg * cos(theta))) + (Fg * cos(theta))
Now, let's substitute the given values:
m = 3.00 kg
g = 9.8 m/s^2 (acceleration due to gravity)
mu = 0.300 (coefficient of static friction)
theta = 35 degrees
Fa = (0.300 * (3.00 * 9.8 * cos(35))) + (3.00 * 9.8 * cos(35))
Calculating this expression will give us the minimum force required to prevent the crate from sliding down the incline.