Two boxes of mass 7.40 and 7.00 kg are in contact and are accelerated across a horizontal surface by an force of 51.49 N applied parallel to the surface and pushing on the lighter box. The magnitude of the frictional force between the surface and the lighter mass is 9.26 N and between the surface and the heavier mass is 15.23 N.
a)Find the acceleration of the two boxes.
b)Find the magnitude of the force exerted by the lighter box on the heavier box in the previous problem.
c)What is the magnitude of the force exerted by the heavier box on the lighter box.
To solve this problem, we can use Newton's second law of motion, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration.
Let's denote the mass of the lighter box as m1 (7.00 kg) and the mass of the heavier box as m2 (7.40 kg). The force applied is F (51.49 N), the magnitude of the frictional force on the lighter box is f1 (9.26 N), and the magnitude of the frictional force on the heavier box is f2 (15.23 N).
a) To find the acceleration of the two boxes, we need to calculate the net force acting on the system and then divide it by the total mass of the system.
The net force (F_net) acting on the system is the force applied minus the frictional force on the lighter mass:
F_net = F - f1
Calculating this:
F_net = 51.49 N - 9.26 N
F_net = 42.23 N
Now, we can use Newton's second law to find the acceleration:
F_net = (m1 + m2) * a
Substituting the values:
42.23 N = (7.00 kg + 7.40 kg) * a
Simplifying:
42.23 N = 14.40 kg * a
Dividing both sides by 14.40 kg:
a = 2.93 m/s^2
Therefore, the acceleration of the two boxes is 2.93 m/s^2.
b) To find the magnitude of the force exerted by the lighter box (m1) on the heavier box (m2), we can use Newton's third law, which states that the forces between two interacting objects are equal in magnitude and opposite in direction.
This means that the force exerted by the lighter box on the heavier box is equal in magnitude to the force exerted by the heavier box on the lighter box.
Therefore, the magnitude of the force exerted by the lighter box on the heavier box is 51.49 N.
c) Similarly, the magnitude of the force exerted by the heavier box on the lighter box is also 51.49 N, as per Newton's third law.
a) To find the acceleration of the two boxes, we need to apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma).
In this case, the net force is the applied force minus the frictional force. The frictional force is opposing the motion, so it is subtracted. Therefore, the net force can be calculated as follows:
Net force = Applied force - Frictional force
= 51.49 N - 9.26 N
= 42.23 N
The net force acting on the system of two boxes is equal to the total mass (m_total) multiplied by the acceleration (a):
F = m_total * a
Since we have two boxes, the total mass (m_total) is the sum of the individual masses:
m_total = m1 + m2
m1 is the mass of the lighter box, which is 7.00 kg, and m2 is the mass of the heavier box, which is 7.40 kg.
Plugging in the values, we have:
42.23 N = (7.00 kg + 7.40 kg) * a
Simplifying the equation:
42.23 N = 14.40 kg * a
Finally, we can solve for the acceleration (a):
a = 42.23 N / 14.40 kg
a ≈ 2.93 m/s²
Therefore, the acceleration of the two boxes is approximately 2.93 m/s².
b) To find the magnitude of the force exerted by the lighter box on the heavier box, we can use Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
The force exerted by the lighter box on the heavier box is equal in magnitude but opposite in direction to the force exerted by the heavier box on the lighter box. So, we need to find the force exerted by the heavier box on the lighter box.
The force exerted by the heavier box on the lighter box can be calculated using Newton's second law:
Force = mass * acceleration
We already know the mass of the heavier box (7.40 kg) and the acceleration (2.93 m/s²) from the previous part.
Force = 7.40 kg * 2.93 m/s²
Force ≈ 21.62 N
Therefore, the magnitude of the force exerted by the lighter box on the heavier box is approximately 21.62 N.
c) As mentioned earlier, the force exerted by the heavier box on the lighter box is equal in magnitude but opposite in direction to the force exerted by the lighter box on the heavier box. Therefore, the magnitude of the force exerted by the heavier box on the lighter box is also approximately 21.62 N.