A box is on a flat plane and has a force of 16 N applied downward to it at an angle of 55 degrees. The box has a mass of 28 kg.
What is the minimum horizontal force F needed to make the box start moving in (Figure 1)? The coefficients of kinetic and static friction between the box and the floor are 0.25 and 0.41, respectively.
Fap = 16N[55o], Downward.
M*g = 28*9.8 = 274.4 N. = Wt. of box.
Fn = Mg + Fap*Sin55 = 274.4 + 16*sin55 = 287.5 N.
Fs = u*Fn = 0.41 * 287.5 = 117.9 N. = Force of static friction.
F+Fap*Cos55-Fs = M*a.
F+16*Cos55-117.9 = 28*0
F + 9.18 - 117.9 = 0
F = 108.7 N.
To find the minimum horizontal force required to make the box start moving, we need to consider the forces acting on the box and the conditions for it to move.
Let's analyze the forces acting on the box:
1. Force applied downward: The force applied to the box is 16 N at an angle of 55 degrees downward with respect to the horizontal plane. We can resolve this force into two components: a vertical component and a horizontal component.
- Vertical component: This force component acts against the normal force and is given by F_vertical = 16 N * sin(55 degrees).
- Horizontal component: This force component acts parallel to the surface and contributes to the force needed to make the box move. It is given by F_horizontal = 16 N * cos(55 degrees).
2. Normal force: The normal force is the force exerted by the surface on the box perpendicular to the contact surface. Its magnitude is equal to the weight of the box, which is given by the product of mass and acceleration due to gravity: N = m * g.
3. Frictional forces: There are two types of friction we need to consider: static friction and kinetic friction.
- Static friction: Static friction is the force that prevents the box from initially moving when an external force is applied. The maximum static friction is given by the product of the normal force and the coefficient of static friction: F_static_friction = μ_static * N.
- Kinetic friction: Kinetic friction is the force that opposes the motion of the box once it starts moving. The magnitude of kinetic friction is given by the product of the normal force and the coefficient of kinetic friction: F_kinetic_friction = μ_kinetic * N.
For the box to start moving, the applied horizontal force must overcome the static friction force. Once the box is in motion, the required force reduces to the kinetic friction force.
Therefore, the minimum horizontal force F required to make the box start moving is equal to the maximum static friction force F_static_friction.
Substituting the given values:
m = 28 kg (mass of the box)
μ_static = 0.41 (coefficient of static friction)
g = 9.8 m/s^2 (acceleration due to gravity)
N = m * g = 28 kg * 9.8 m/s^2
F_static_friction = μ_static * N = 0.41 * (28 kg * 9.8 m/s^2)
Finally, the minimum horizontal force F needed to make the box start moving is equal to F_static_friction.
Calculating the values will give you the answer.