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. Find the acceleration of the two boxes.
Find the magnitude of the force exerted by the lighter box on the heavier box.
To find the acceleration of the two boxes, we can use Newton's second law of motion. This law states that the net force acting on an object is equal to the product of its mass and acceleration.
For the lighter box:
Net force = Applied force - Frictional force
Net force = 51.49 N - 9.26 N
Net force = 42.23 N
For the heavier box:
Net force = Applied force - Frictional force
Net force = 51.49 N - 15.23 N
Net force = 36.26 N
Now we can calculate the acceleration of each box using Newton's second law:
For the lighter box:
Net force = mass * acceleration
42.23 N = 7.00 kg * acceleration
Therefore, the acceleration of the lighter box is:
acceleration = 42.23 N / 7.00 kg = 6.03 m/s²
For the heavier box:
Net force = mass * acceleration
36.26 N = 7.40 kg * acceleration
Therefore, the acceleration of the heavier box is:
acceleration = 36.26 N / 7.40 kg = 4.91 m/s²
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.
Therefore, the magnitude of the force exerted by the lighter box on the heavier box is equal to the magnitude of the force exerted by the heavier box on the lighter box. In this case, it is equal to the frictional force between the two boxes.
So, the magnitude of the force exerted by the lighter box on the heavier box is:
Force = 15.23 N
To find the acceleration of the two boxes, we can use Newton's second law of motion. The formula for Newton's second law is:
F = m*a
Where:
F is the net force applied to an object,
m is the mass of the object, and
a is the acceleration of the object.
For the lighter box:
The net force acting on the lighter box is the applied force minus the frictional force. So we have:
F_net_lighter = F_applied - F_friction_lighter
Substituting the given values:
F_net_lighter = 51.49 N - 9.26 N = 42.23 N
We can rearrange Newton's second law equation to solve for acceleration:
a_lighter = F_net_lighter / m_lighter
Substituting the known values:
a_lighter = 42.23 N / 7.00 kg ≈ 6.04 m/s²
For the heavier box:
The net force acting on the heavier box is the frictional force on the lighter box minus the frictional force on the heavier box. So we have:
F_net_heavier = F_friction_lighter - F_friction_heavier
Substituting the given values:
F_net_heavier = 9.26 N - 15.23 N = -5.97 N
Since the net force on the heavier box is negative, it means the force exerted by the lighter box on the heavier box must be in the opposite direction. Therefore, the magnitude of the force exerted by the lighter box on the heavier box would be:
|F_lighter_on_heavier| = |-5.97 N| = 5.97 N
Thus, the magnitude of the force exerted by the lighter box on the heavier box is 5.97 N.