Q25E - University Physics with Modern Physics

Question

Crates A and B sit at rest side by side on a frictionless horizontal surface. They have masses ${\mathbf{m}}_{\mathbf{A}}$ and ${\mathbf{m}}_{\mathbf{B}}$ respectively. When a horizontal force  $\stackrel{\rightharpoonup }{\mathbf{F}}$ is applied to crate A, the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate A and for crate B. Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of $\overrightarrow{\mathbf{F}}$ is less than the total weight of the two crates, will it cause the crates to move? Explain.

Step-by-Step Solution

Answer

(a) The free-body diagram for crate A and for crate B is,

                 

The normal reaction force ${\mathrm{N}}_1$ on the crate A and the crate B are the action-reaction pairs, ${\mathrm{R}}_{\mathrm{A}}$ and ${\mathrm{R}}_{\mathrm{B}}$ are the normal reaction of ground against the respective weight of crates are action-reaction pairs and also the weight force of the crate A and the crate B  are action-reaction pairs.

(b) The crate will move in the right direction under the action of the horizontal force as the surface is frictionless and there is zero net force in the vertical direction.

Step 1: Significance of linear acceleration:

Whenever an object of specific mass moves in a particular direction under the action of an external force, then there would be a linear acceleration exists. The concept of Newton’s law of motion helps to obtain the value of the acceleration.

Step 2: Identification of the given data

The given data can be listed below as,

The mass of a crate A is ${\mathrm{m}}_{\mathrm{A}}$.

The mass of a crate B is ${\mathrm{m}}_{\mathrm{B}}$.

Step 3: (a) Draw the free-body diagram for crate A and for crate B:

The free-body diagram of crate A and crate B is given below.

Here, ${\mathrm{N}}_1$ is the normal reaction force between the crate A and crate B due to action and reaction of the two crates, ${\mathrm{R}}_{\mathrm{A}}$ and ${\mathrm{R}}_{\mathrm{B}}$ are the normal reaction of ground against the respective weight of crates and ${\mathrm{m}}_{\mathrm{A}}g$, ${\mathrm{m}}_{\mathrm{B}}g$ are the weight force of the crate A and the crate B respectively.

The normal reaction force ${\mathrm{N}}_1$ on the crate A and ${\mathrm{N}}_1$ on the crate B are according to the third-law action-reaction pairs and also the weight force of the crate A and the crate B  are according to the third-law action-reaction pairs.

Thus, the normal reaction force ${\mathrm{N}}_1$ on the crate A and on the crate B are action-reaction pairs, ${\mathrm{R}}_{\mathrm{A}}$ and ${\mathrm{R}}_{\mathrm{B}}$ are the normal reaction of ground against the respective weight of crates are action-reaction pairs and also the weight force of the crate A and the crate B  are action-reaction pairs.

Step 4: (b) Determination of the possibility of the motion of the crate system

The free-body diagram of the composite system of crate A and crate B is given below.

                                       

                                 

Here, $\mathrm{W}$ is the total weight force of the crate A and crate B, $({\mathrm{R}}_{\mathrm{A}}+{\mathrm{R}}_{\mathrm{B}})$ is the total reaction of the both crates.

From the free-body diagram of the composite system, there is only one horizontal force towards the right direction, but the net force in the vertical direction is equal to zero. So, the object will move in a horizontal direction irrespective of the weight of the crate as the surface is frictionless.

If the magnitude of $\mathrm{F}$ is less than the total weight of the two crates, then the crate will move in the horizontal right direction.

Thus, the crate will move in the right direction under the action of the horizontal force as the surface is frictionless and there is zero net force in the vertical direction.