When a horizontal force of 4.5 newtons acts on a block on a resistance-free surface, it produces an acceleration of 2.5 m/s squared. Suppose a second 4.0 kg block is dropped onto the first. What is the magnitude of the acceleration of the combination if the same force continues to act? Assume that the second block does not slide on the first block.
Well, well, well! Looks like we have a classic physics problem here. Let's have some fun with it!
If we're dropping a 4.0 kg block onto the first block, we need to consider the total mass now. The mass of the first block is 4.0 kg, and the mass of the second block is...uh...4.0 kg as well. So we have a grand total of 8.0 kg to deal with, double the fun!
Now, we know that the horizontal force is still 4.5 newtons, and the acceleration of the first block alone is 2.5 m/s². But what about the combined acceleration?
To find that, we can use Newton's second law, which states that force equals mass times acceleration (F = ma). So, if we have a total mass of 8.0 kg, we can rearrange the equation to find the combined acceleration.
F = ma
4.5 N = 8.0 kg * a
Dividing both sides of the equation by 8.0 kg, we get:
a = 4.5 N / 8.0 kg = 0.5625 m/s²
Ta-da! The magnitude of the acceleration of the combination is 0.5625 m/s². And with that, the blocks are ready to rock and roll!
To find the magnitude of the acceleration of the combination when the second block is dropped onto the first, we can use the concept of Newton's Second Law of Motion.
Newton's Second Law states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. Mathematically, it can be represented as:
F = ma
Where:
F = net force applied to the object
m = mass of the object
a = acceleration of the object
In this case, when a horizontal force of 4.5 newtons is applied to the first block, it produces an acceleration of 2.5 m/s^2. We can write the equation as:
4.5 = (m1 + m2) * a
where:
m1 = mass of the first block (given mass)
m2 = mass of the second block (given mass)
a = acceleration of the combination
Since the second block is dropped onto the first block, we can assume that there is no additional force acting on the system other than the horizontal force of 4.5 newtons.
Given that the mass of the first block (m1) is 4.0 kg and the mass of the second block (m2) is 4.0 kg, we substitute these values into the equation:
4.5 = (4.0 + 4.0) * a
Simplifying further:
4.5 = 8.0 * a
We can solve for the acceleration (a) by dividing both sides of the equation by 8.0:
a = 4.5 / 8.0
a ≈ 0.5625 m/s^2
Therefore, the magnitude of the acceleration of the combination, when the same force continues to act after the second block is dropped onto the first, is approximately 0.5625 m/s^2.
To find the magnitude of the acceleration of the combination when the second block is dropped onto the first, we need to consider the forces acting on the system.
First, let's calculate the mass of the combined system. Since a second block of mass 4.0 kg is dropped onto the first block, the combined mass will be the sum of the individual masses:
Total mass (m) = Mass of first block + Mass of second block
= 4.0 kg + 4.0 kg
= 8.0 kg
Now, let's consider the forces acting on the system. The horizontal force of 4.5 N continues to act on the combined system. In addition to that, we have the gravitational force acting on both blocks.
The gravitational force acting on the first block is given by:
Force (F) = mass (m) × acceleration due to gravity (g)
= 4.0 kg × 9.8 m/s^2 (approximately, acceleration due to gravity)
= 39.2 N
Since the second block is dropped onto the first block, the gravitational force acting on the second block is also 39.2 N.
Now, let's calculate the net force acting on the system. The horizontal force of 4.5 N acts in one direction, and the gravitational forces on both blocks act in the opposite direction. The net force on the system can be calculated as the difference between the applied force and the gravitational forces:
Net Force (F_net) = Applied force - Gravitational forces
= 4.5 N - (39.2 N + 39.2 N)
= 4.5 N - 78.4 N
= -73.9 N
Notice that the net force is negative because the gravitational forces are greater than the applied force, causing deceleration or negative acceleration.
Now, we can calculate the acceleration of the combined system using Newton's second law of motion:
F_net = m × a
Rearranging the formula to solve for acceleration (a):
a = F_net / m
Plugging in the values:
a = -73.9 N / 8.0 kg
a ≈ -9.24 m/s^2
The magnitude of the acceleration of the combination is approximately 9.24 m/s^2. Notice that the negative sign indicates that the system is decelerating or slowing down.
first block=F=ma , but this same force acts after the second block is dropped.
F=(m+M)a' = ma
solve for a'