Mechanics Problems max, instant. v, etc.

[Shoelace Thm.]

Homework Statement

A 10 kg crate is pulled with a force F_A at an angle \theta to accelerate the crate at 0.9 m/s^2. The coefficient of friction between the floor and the crate is 0.45. Derive an expression for the angle that the crate be pulled so that the applied force is a minimum. Calculate the value of the angle and minimum force.

2. A robot working in a nuclear power plant moves along a straight track. If it experiences a force

F(t) = -F_i[1+\frac{(4.0t-2.0\tau)}{\tau}]
where \tau is a constant with units of time, what is the instantaneous speed of the robot at the very end of the interval 0 \le t \le \tau? At t=0 sec, v=0m/s.

Block 1 of mass m_1 is placed on block 2 of mass m_2, which is then placed on a table. A string connecting block 2 of mass M passes over a pulley attached to one end of the table (it is a normal table e.g., the edges meet perpendicularly). The mass and friction of the pulley are negligible. The coefficients of friction between blocks 1 and 2 are nonzero and are given by \mu _{s1} (static), \mu_{k1} (kinetic). The coefficients of friction between block 2 and the tabletop are \mu _{s2} (static), \mu_{k2} (kinetic). Express your answers in terms of masses, coefficients of friction, and g, the acceleration due to gravity.
(a) Suppose that the value of M is small enought that the blocks remain at rest when released. For each of the following forces, determine the magnitude of the force and draw a vector on the block (you can explicitly state it if you want, but I would assume you would know the free-body to determine the force) to indicate the direction of the force (if it is nonzero): (1) The normal force N_1 exerted on block 1 by block 2. (2) The friction force f_1 exerted on block 1 by block 2. (3) The force F_t exerted on block 2 by the string. (4) The normal force N_2 exerted on block 2 by the tabletop. (5) The friction force f_2 exerted on block 2 by the tabletop.
(b) Determine the largest value of M for which the blocks can remain at rest.
(c) Now suppose that M is large enought that the hanging block descends when the blocks are released. Assume that blocks 1 and 2 are moving as a unit (no slippage). Determine the magnitude a of their acceleration.
(d) Now suppose that M is large enough that as the hanging block descends, block 1 is slipping on block 2. Determine each of the following: (1) The magnitude a_1 of the acceleration of block 1. (2) The magnitude a_2 of the acceleration of block 2.

Homework Equations

The Attempt at a Solution

For one, the minimization of the quantity (using inequality bounds?) or differentiation if it works, just please carry it through, especially the former (I have the quantity for \theta).

For two, just start the integration please of the quantity/[m], a few steps please, enough for me to carry through.

I got three, but I found it an interesting problem, especially the bounds for slippage. You can post solutions, or comments if you wish.

 

[Ambidext]

Is the F_A at an angle against gravity? If what I understand is correct, then:

First fine the kinetic friction using:
f_k = 0.45(mg - F_A sin theta )

And then form an equation that involves F_A and f_k. Divide the equation into horizontal and vertical components first.

 

[Shoelace Thm.]

I got 45 degrees, so can you confirm that? Also can you start #2 please?