A block of mass M1positioned on a slope (at an angle ) is connected by
means of an inextensible wire, of negligible mass, to another mass, m2,
suspended from a frictionless pulley. The coefficients of static and dynamic
friction between the slope and M1are 0.3 and 0.2 respectively
(a) Determine the mass of m2for M1to be about to move downthe slope.
(b) Determine the mass of m2for M1to be about to move upthe slope.
(c) Once M1is moving up the slope, determine its acceleration.
(d) After m2falls through a distance of 30 cm, what is the velocity of M1?
it is not clear to me if M2 is tending to pull it up or down, as the position of the pulley is not stated.
is there any way i can get ur email so i can send the pic of the question
To solve this problem, we can break it down into smaller steps and use the principles of Newton's laws of motion.
(a) To determine the mass of m2 for M1 to be about to move down the slope, we need to consider the forces acting on M1. There are two forces in play: the gravitational force pulling M1 downwards and the frictional force pushing M1 up the slope.
The force due to gravity can be calculated as F_gravity = M1 * g * sin(θ), where g is the acceleration due to gravity and θ is the angle of the slope.
The frictional force can be calculated as F_friction = coefficient of static friction * normal force, where the normal force is the force perpendicular to the slope. The normal force is given by N = M1 * g * cos(θ). Therefore, the frictional force is F_friction = coefficient of static friction * M1 * g * cos(θ).
For M1 to be about to move, the force due to gravity pulling it down must be greater than or equal to the frictional force pushing it up. Therefore, we can set up the following equation: M1 * g * sin(θ) >= coefficient of static friction * M1 * g * cos(θ).
(b) To determine the mass of m2 for M1 to be about to move up the slope, we need to consider the forces acting on M1 in the opposite direction. Now, the force due to gravity is still pulling M1 downwards, but the frictional force is now acting in the same direction as the force due to gravity. The frictional force can be calculated as F_friction = coefficient of dynamic friction * normal force. Again, the normal force N = M1 * g * cos(θ). Therefore, the frictional force is F_friction = coefficient of dynamic friction * M1 * g * cos(θ). For M1 to be about to move up, the force due to gravity pulling it down must be equal to the frictional force pushing it up (as there is no longer a static friction force acting in the opposite direction). Thus, we can set up the equation: M1 * g * sin(θ) = coefficient of dynamic friction * M1 * g * cos(θ).
(c) Once M1 is moving up the slope, its acceleration can be determined using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration, or ΣF = M1 * a. The net force acting on M1 can be calculated as the difference between the force due to gravity pulling it down and the force of kinetic friction pushing it up. Therefore, we can set up the equation M1 * g * sin(θ) - coefficient of dynamic friction * M1 * g * cos(θ) = M1 * a, and solve for a.
(d) After m2 falls through a distance of 30 cm, we can calculate the velocity of M1 using the principle of conservation of mechanical energy. The potential energy lost by m2 can be converted into the kinetic energy of M1. Therefore, we can use the equation m2 * g * h = (1/2) * M1 * v^2, where m2 is the mass of m2, g is the acceleration due to gravity, h is the height m2 falls through (in this case, 30 cm or 0.3 m), M1 is the mass of M1, and v is the velocity of M1.