The drawing shows a large cube (mass = 48 kg) being accelerated across a horizontal frictionless surface by a horizontal force . A small cube (mass = 2.1 kg) is in contact with the front surface of the large cube and will slide downward unless is sufficiently large. The coefficient of static friction between the cubes is 0.71. What is the smallest magnitude that can have in order to keep the small cube from sliding downward?
52.2
To find the smallest magnitude that force can have in order to prevent the small cube from sliding downward, we need to consider the forces acting on the small cube.
1. Gravity: The force due to gravity acting on the small cube is given by the formula F_gravity = m * g, where m is the mass of the small cube and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (2.1 kg) * (9.8 m/s^2) = 20.58 N
2. Friction: The static friction opposing the motion between the large and small cubes can prevent the small cube from sliding downward. The maximum static friction force is given by the formula F_friction_max = μ * N, where μ is the coefficient of static friction and N is the normal force.
In this case, the normal force acting on the small cube is equal to the weight of the small cube, which is equal to the force due to gravity.
N = (2.1 kg) * (9.8 m/s^2) = 20.58 N
Therefore, the maximum static friction force can be calculated as:
F_friction_max = (0.71) * (20.58 N) = 14.61 N
Since the force due to gravity acting on the small cube (20.58 N) is greater than the maximum static friction force (14.61 N), the small cube will start to slide downward unless another force is applied.
To prevent the small cube from sliding downward, the horizontal force applied to the large cube must be equal to the maximum static friction force acting on the small cube. Therefore, the smallest magnitude that the force can have in order to keep the small cube from sliding downward is 14.61 N.