# Solving Package A's Motion w/ Newton's 3rd Law

• FancyNut
In summary, Newton's 3rd law states that the sum of the action and reaction forces on an object is equal to the object's mass multiplied by the coefficient of kinetic friction.
FancyNut
Newton's 3rd law...

I'm having trouble with this problem...

Two packages at UPS start sliding down the 20 degree ramp shown in the figure. Package A has a mass of 3.50 kg and a coefficient of kinetic friction of 0.200. Package B has a mass of 8.50 kg and a coefficient of kinetic friction of 0.130.

How long does it take package A to reach the bottom?

Picture is attached...

Now since both have different masses and friction coefficients I'm thinking they each take a different amount of time to reach the bottom. However because at the beginning they're an action/reaction pair I ADDED the x-component of weight for B to the forces acting on A when calculating the acceleration... then used kinematics :

$$x_f = x_i + v_i t + 1/2 a (t)^2$$

where x final is 2 meters and initial x and v are zero and acceleration is the one I got from force analysis...

I've been at this problem for a few hours-- going back to it after every while and still have no idea on what to do. btw does it have calculus? It could be that. My professor puts in one or two questions with calculus in HW/midterm preperation questions but we never actually study physics problems with calculus. :(

thanks in advance for any help. :)

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ok now I think I did a big mistake...

I rewrote the free-body diagrams for each block and labeled the action/reaction force as F_a on b which is positve and F_b on a which is negative... now I have two equations with two unknows: the acceleration which is the same at the start and the action/reaction force. I can't subtract second equation from the first because that won't cancel out the action/reaction force since they have opposite signs...

stuck again. :(

First assume that the blocks do not interfere. After drawing the freebody diagrams and writing the equations you will arrive at
$$a_1 = g\sin \alpha - \mu_{1}g\cos \alpha$$
$$a_2 = g\sin \alpha - \mu_{2}g\cos \alpha$$

Because $$\mu_{1} > \mu_{2}$$
we have $$a_1 < a_2$$

The assumption that the bodies behave seperately is not correct.
Therefore both the bodies have the same acceleration "a"

Then once again drawing the corresponding free body diagrams which also involve the force of contact between the blocks, we arrive at (Correct me if am wrong)

$$a= \frac {(m_1 + m_{2})g\sin \alpha - g\cos \alpha(m_{1} \mu_{1} + m_{2} \mu{2})}{(m_{1} + m_{2})}$$

After this it is easy to calculate the time.

More importantly the concept behind this problem is that both the blocks move as one unit. Now take your co-ordinate system as perpendicular and parallel to the plane. (This is beacuse the distance to be covered is given parallel to the plane). Also while drawing the free body diagrams include the force exerted by block A on block B and vice versa. Notice how this force cancels when you add the two equations?. If you can't subtract... just add :)

Thanks for the help!

I learned a lot from this problem thanks to you.

## 1. What is Newton's 3rd Law?

Newton's 3rd Law states that for every action, there is an equal and opposite reaction. This means that when an object exerts a force on another object, the second object will exert an equal and opposite force back on the first object.

## 2. How does Newton's 3rd Law apply to solving Package A's motion?

In order to solve Package A's motion, we must consider the forces acting on the package and apply Newton's 3rd Law. This means that for every force acting on Package A, there is an equal and opposite force acting on another object, such as the surface it is resting on or the hand holding it.

## 3. What are the steps for solving Package A's motion using Newton's 3rd Law?

The steps for solving Package A's motion with Newton's 3rd Law are as follows:

1. Identify all the forces acting on Package A.

2. Determine the direction and magnitude of each of these forces.

3. Use Newton's 3rd Law to find the equal and opposite forces acting on other objects.

4. Use the net force acting on Package A to calculate its acceleration using Newton's 2nd Law (F=ma).

5. Use the acceleration and initial conditions to calculate the displacement and velocity of Package A.

## 4. Can Newton's 3rd Law be applied to all types of motion?

Yes, Newton's 3rd Law can be applied to all types of motion, including linear, circular, and rotational motion. It is a fundamental principle of physics that governs the interaction of objects in the universe.

## 5. How does understanding Newton's 3rd Law benefit scientists and engineers?

Understanding Newton's 3rd Law allows scientists and engineers to accurately predict and manipulate the motion of objects. This knowledge is essential for designing and creating structures, machines, and technology that function effectively and safely. It also allows for the study and exploration of the natural world and the laws that govern it.

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