The drawing shows a large cube (mass=42 kg) being accelerated across a horizontal table by a horizontal force P. The coefficient of kinetic friction between the table and large cube is 0.24. A small cube (mass= 5.0 kg) is in contact with the front surface of the large cube and will slide downward unless P is sufficiently large. The coefficient of static friction between the cubes is .71. What is the smallest magnitude that P can have in order to keep the small cube from sliding down?
To determine the minimum magnitude of force (P) required to keep the small cube from sliding down, we need to consider the forces acting on both cubes.
Let's analyze the forces acting on the small cube first:
1. Weight force (Fg1): The weight force acts vertically downward and can be calculated by multiplying the mass of the small cube (5.0 kg) by the acceleration due to gravity (9.8 m/s^2). Fg1 = 5.0 kg * 9.8 m/s^2 = 49 N.
Next, let's analyze the forces acting on the large cube:
1. Weight force (Fg2): The weight force acts vertically downward and can be calculated by multiplying the mass of the large cube (42 kg) by the acceleration due to gravity (9.8 m/s^2). Fg2 = 42 kg * 9.8 m/s^2 = 411.6 N.
2. Normal force (Fn2): The normal force acts perpendicular to the surface of contact (horizontal table) and is equal in magnitude to the weight force. Therefore, Fn2 = Fg2 = 411.6 N.
3. Friction force (Ff2): The friction force opposes the motion of the large cube and acts parallel to the surface of contact. The magnitude of the friction force can be determined using the equation Ff2 = μk * Fn2, where μk is the coefficient of kinetic friction (0.24 in this case). Therefore, Ff2 = 0.24 * 411.6 N = 98.78 N.
The force P must be large enough to overcome the force of static friction between the cubes (Ffs), which prevents the small cube from sliding down. The force of static friction can be calculated using the equation Ffs = μs * Fn2, where μs is the coefficient of static friction (0.71 in this case). Therefore, Ffs = 0.71 * 411.6 N = 292.65 N.
To prevent the small cube from sliding down, the force P must be equal to or greater than the force of static friction (Ffs). Therefore, the smallest magnitude that P can have is 292.65 N.