QP4

The system of blocks and pulleys shown above is being pulled to the right with a force F. This force is adjusted so that block M₁ neither falls nor rises, i.e., the y component of its acceleration vanishes.

All surfaces are frictionless, the pulley is massless and frictionless, and the strings (dotted lines) are massless and inextensible. Strings are marked with their tensions and blocks are marked with their masses. All surfaces shown to be in contact remain in contact at all times; in particular block M₄ does not "tip over". The string connecting blocks M₃ and M₄ makes an angle with the horizontal.

(3 points) (a) Draw free body diagrams for all four masses, with all forces clearly shown and labeled.

(2 points) (b) Write Newton's second law for blocks M₁ and M₂, in the x and y directions (i.e., 4 equations).

(2 points) (c) What acceleration a₃ must block M₃ have? (Recall that block M₁ neither falls nor rises.) Give your answer in terms of the masses M₁, M₂, M₃, M₄ and gravitational acceleration g.

(1 point) (d) What is the magnitude of the force F?

(1 point) (e) What is the normal force N₄ between the floor and block M₄? You may leave your answer in terms of the masses and a₃.

(1 point) (f) What is the force F_p (magnitude and direction) of the pulley on the string? You may leave your answer in terms of the masses and a₃.