Mechanics Problems max, instant. v, etc.

In summary, The conversation discusses various scenarios involving forces, friction, and acceleration in different systems. It deals with determining the minimum angle for pulling a crate, calculating the instantaneous speed of a robot, and analyzing the forces acting on blocks on a table with different coefficients of friction. It also involves finding the largest value of a variable for a system to remain at rest and calculating the acceleration when the system is in motion.
  • #1
Shoelace Thm.
60
0

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.
 
Last edited:
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  • #2
Is the FA at an angle against gravity? If what I understand is correct, then:

First fine the kinetic friction using:
fk = 0.45(mg - FA sin theta )

And then form an equation that involves FA and fk. Divide the equation into horizontal and vertical components first.
 
  • #3
I got 45 degrees, so can you confirm that? Also can you start #2 please?
 

Related to Mechanics Problems max, instant. v, etc.

1. What is the maximum velocity in mechanics problems?

The maximum velocity in mechanics problems refers to the highest speed that an object reaches during its motion. This can be calculated using the equation: V=U+at, where V is the final velocity, U is the initial velocity, a is the acceleration, and t is the time period.

2. How do you calculate instant velocity in mechanics problems?

Instant velocity, also known as instantaneous velocity, is the velocity of an object at a particular moment in time. It can be calculated using the equation: V=lim Δt→0 Δx/Δt, where V is the instant velocity, Δx is the displacement, and Δt is the time interval.

3. What are some common types of mechanics problems?

Some common types of mechanics problems include kinematics problems, which involve the motion of objects without considering the forces that cause the motion, and dynamics problems, which involve the relationship between forces and motion. Other types include statics problems, which deal with objects at rest, and fluid mechanics problems, which involve the study of fluids in motion.

4. How do you approach solving mechanics problems?

The most important step in solving mechanics problems is to clearly understand the given information and what is being asked. Then, choose the appropriate equations and substitute the known values to solve for the unknown quantity. It is also helpful to draw diagrams and use units consistently throughout the problem.

5. What are some common mistakes to avoid in mechanics problems?

Some common mistakes to avoid in mechanics problems include using incorrect units, not considering all the forces acting on an object, and not paying attention to the direction of motion. It is also important to double check calculations and make sure they are consistent with the given information.

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