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?
Please show the steps
To find the smallest magnitude of force P required to keep the small cube from sliding down, we need to determine the maximum force of static friction between the cubes.
First, let's calculate the weight of the small cube:
Weight (W₁) = mass (m₁) × acceleration due to gravity (g)
W₁ = 5.0 kg × 9.8 m/s²
W₁ = 49 N
Next, we need to calculate the weight of the large cube:
Weight (W₂) = mass (m₂) × acceleration due to gravity (g)
W₂ = 42 kg × 9.8 m/s²
W₂ = 411.6 N
The force of static friction between two surfaces is given by:
Force of static friction (F_static) = coefficient of static friction (μ_static) × normal force (N)
To calculate the normal force between the cubes, we need to determine the force exerted by the large cube on the small cube:
Force exerted by large cube on small cube (F_N) = Weight of small cube (W₁)
F_N = 49 N
Now, we can calculate the maximum force of static friction between the cubes:
F_static = coefficient of static friction (μ_static) × normal force (N)
F_static = 0.71 × 49 N
F_static = 34.79 N
Since P needs to be large enough to counteract the force of static friction, the smallest magnitude that P can have to prevent the small cube from sliding down is 34.79 N.