Determine the acceleration that the cart, shown in the figure, must have so that the block does not fall. Assume g for the acceleration due to gravity and μ for the coefficient of friction between the block and cart.

The masses m and M are connected, by two ropes A and B, on an inclined plane elevated, as in the figure. Neglecting the masses of the ropes, and the friction in the pulleys, it is given the inclination angle of the plane equal to θ and the acceleration due to gravity g, determine:
a) The acceleration of the system, knowing that the mass M is descending the plane;
b) The difference between tensions forces T_A and T_B.

In the system shown in the figure, p₁ is a mobile pulley, p₂ is a fixed pulley, the weight of block B is 2000 N, and the angle of the inclined plane is equal to 30°. Determine which should be the weight of block A so that block B has a speed of 20 m/s after a path of 40 m in the upward direction. Neglect the masses of the ropes and the pulleys. Neglect the friction between the ropes and the pulleys and between block B and the plane. Assume g = 10 m/s².

On an inclined plane of 30° to the horizontal, it slides without friction a mass m₁ attached to another mass m₂. The system is released from the rest, the mass m₂ rises 250 m in 20 s. Calculate the ratio m₁/m₂. Assume g = 10 m/s².

In the system of the figure, the masses of A, B, and C are 10 kg, 20 kg, and 5 kg, and sin θ = 0.8. Neglecting the friction forces, calculate the acceleration of the system and the magnitude of the tension forces on the ropes. Assume g = 10 m/s².

In the figure, the coefficient of friction between blocks A, B, and the plane on which they slide is 0.2. The masses of A, B, and C are equal to 100 kg, 50 kg, and 50 kg. Determine the acceleration of each block and the tension force on the rope. Assume g = 10 m/s² and 2a_C = a_A+a_B, a_A, a_B and a_C are the accelerations of blocks A, B, and C. The rope and the pulleys are frictionless and lightweight.

A cart with mass M is attached by a rope to a load with mass m. In the initial instant, the cart has speed v₀ and moves to the left in a horizontal plane. Determine:
a) The interval of time until the cart stop;
b) The displacement until the cart stop.
Considering the cord inextensible and negligible mass, there is no friction in the horizontal plane and on the pulley, and assume the acceleration due to gravity is equal to g.

What horizontal force should be constantly applied to the mass M = 21 kg so that the mass m₁ = 5 kg does not move relative to the mass m₂ = 4 kg? Neglect the friction and assume g = 10 m/s².