A block of mass 3m is placed on a frictionless horizontal surface, and a second block of mass m is placed on top of the first block. The surfaces of the blocks are rough. A constant force of magnitude F is applied to the first block.
a) Identify the horizontal force that causes the block of mass m to accelerate.
b) Assume that the upper block does not slip on the lower block, and find the acceleration of each block in terms of m and F.
a) Well, the horizontal force that causes the block of mass m to accelerate is none other than the force itself, F. It's a classic case of "what goes around comes around." That force is no joke!
b) Now, let's put on our clown shoes and calculate the acceleration of each block. Since the upper block doesn't slip on the lower block, the force of friction between them will be the same as the applied force F.
For the lower block:
The net force acting on the lower block is F - f, where f is the force of friction between the blocks. Since the surface is rough, the friction force can be calculated as f = μmg, where μ is the coefficient of friction and g is the acceleration due to gravity. Thus, the net force on the lower block is F - μmg.
According to Newton's second law (F = ma), the net force is also equal to the product of mass and acceleration. Hence, we have F - μmg = (3m)a (since the mass of the lower block is 3m).
For the upper block:
The force of friction between the upper and lower blocks will be f as well. As per Newton's second law, F - f = ma (where a is the acceleration of the upper block, which is also the same for the lower block since they're connected).
Now, we can solve this clownish puzzle using some fancy algebra. Rearranging the equations, we get:
For the lower block: F = (3m)a + μmg
For the upper block: F = ma + f
Since the forces are the same, we can equate the two equations:
(3m)a + μmg = ma + f
Now we substitute the friction force: f = μmg
(3m)a + μmg = ma + μmg
Now, let's simplify it even further:
(3m)a = ma
The masses cancel out! It's like they vanished into thin air (but don't worry, they're still there). So, no matter how big or small the mass m is, the acceleration of both blocks will be the same.
To answer these questions, we need to consider the forces acting on each block and apply Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
a) To identify the horizontal force that causes the block of mass m to accelerate, we need to consider the forces acting on the block.
On the block of mass m, there are two forces acting horizontally: the applied force F and the frictional force between the two blocks. Since we are assuming the upper block does not slip on the lower block, the frictional force will also act on the block of mass m.
Now, the frictional force between two objects depends on the coefficient of friction (μ) and the normal force (N) between them. In this case, the normal force on the block of mass m is equal to its weight, which is mg, where g is the acceleration due to gravity.
The maximum frictional force (F_friction_max) between two surfaces is given by the equation F_friction_max = μN. But we need to find the actual frictional force (F_friction), which is usually less than the maximum value. Typically, this force can be found using the equation F_friction = μN. But in this case, the frictional force will be equal to the total weight of both blocks, since there is no slipping.
So, the horizontal force that causes the block of mass m to accelerate is the applied force F.
b) To find the acceleration of each block in terms of m and F, we can apply Newton's second law separately to each block.
For the block of mass m:
The net force acting on it is the difference between the applied force F and the frictional force (which is the weight of both blocks).
So, the net force (F_net_m) acting on the block of mass m is F - (2m)g, where g is the acceleration due to gravity.
Using Newton's second law, we have:
F_net_m = m * a_m,
where a_m is the acceleration of the block of mass m.
Rearranging the equation, we can find the acceleration of the block of mass m:
a_m = (F - (2m)g) / m.
For the block of mass 3m:
The net force acting on it is the force of friction, which is equal to the total weight of both blocks.
So, the net force (F_net_3m) acting on the block of mass 3m is (2m)g, where g is the acceleration due to gravity.
Using Newton's second law, we have:
F_net_3m = 3m * a_3m,
where a_3m is the acceleration of the block of mass 3m.
Rearranging the equation, we can find the acceleration of the block of mass 3m:
a_3m = (2m)g / (3m).
Therefore, the acceleration of each block in terms of m and F is:
- The block of mass m: a_m = (F - (2m)g) / m.
- The block of mass 3m: a_3m = (2m)g / (3m).