Two wooden blocks, connected by a massless string, are moving at constant velocity over a floor with friction. The smaller of the two blocks is pulled by a horizontal force of 500 N.
Knowing that the mass of the smaller block is 30 kg and the mass of the larger block is 40 kg, find the tension in the string and the coefficient of sliding friction between the blocks and the floor.
Let's begin by drawing a sketch that represents what is happening in the problem.
In the sketch, we draw a horizontal floor with two blocks that are connected by a string sitting on it. Then, we indicate that the smaller of the two blocks is pulled by a horizontal force and that both the blocks are subject to the force of friction. It's also a good idea to write that the two blocks move at constant velocity. The final sketch looks something like this:
Since the string connecting the two blocks is massless, the tension forces exerted by the ends of the string have the same magnitude, which we will indicate with T.
Also, since the two blocks are made of the same material (wood) and they move on the same floor, the coefficient of sliding friction is the same for both blocks, and we will indicate it with μ.
In this problem, we are dealing with two blocks, so it's a good idea to analyze them separately.
Looking carefully at our sketch, we can see that the smaller block is subject to 5 forces:
Here's the free-body diagram of the smaller block:
The larger block, on the other hand, is subject to 4 forces:
Here's the free-body diagram of the larger block:
We know the masses of the two blocks (30 kg and 40 kg), the magnitude of the pulling force (500 N), and that the two blocks are moving at constant velocity.
We need to find the magnitude of the tension in the string (T), and the coefficient of sliding friction (μ).
Since the blocks are moving at constant velocity, they have zero acceleration, so the net force on the smaller block (fnet) is zero, and the net force on the larger block (Fnet) is also zero:
We can find the tension in the string and the coefficient of sliding friction following these steps:
Let's begin with step 1.
To find the components of the net force that acts on the smaller block, we first draw the coordinate axes on the free-body diagram of the smaller block and enumerate the components of all the forces that act on it:
We can now find the x and y components of the net force by adding the x and y components of all the forces:
x:
y:
And we replace fnetx and fnety with 0:
Additionally, we know that the magnitude of the friction force ff is the product between the coefficient of friction μ and the magnitude of the normal force n:
Therefore, Eq. (1) becomes:
Next, let's solve Eq. (2) for n:
And replace n with mg in Eq. (3):
This final equation has 2 unknowns (T and μ).
Let's now repeat the same process on the larger block in order to find the second equation.
Again, we draw the coordinate axes on the free-body diagram and determine the x and y components of each force:
Then, we calculate the x and y components of the net force Fnet:
x:
y:
Again, because Fnet is zero, the components are zero as well, so we can replace them in the above equations with 0:
We can write the magnitude of the friction force as:
Therefore, Eq. (5) becomes:
Next, let's solve Eq. (6) for N:
And swap N with Mg in Eq. (7):
This is the final equation for the larger block.
Thus, the two final equations are Eq. (4) and Eq. (8):
These are two equations with two unknowns (T and μ), so we can solve them.
Let's solve Eq. (8) for T:
And replace T with μMg in Eq. (4):
Now, let's solve this equation for μ:
| μ = | F |
| g (M + m) |
| μ = | 500 N |
| (9.8 N/kg) (40 kg + 30 kg) |
| μ = | 500 N |
| (9.8 N/kg) (70 kg) |
Lastly, we can get T from Eq. (9):
Therefore, the tension in the string is 2.9 × 102 N, and the coefficient of sliding friction is 0.73.