To solve this problem, we need to consider the forces acting on the system and how they affect the acceleration. Let's break down the steps:Step 1: Identify the forces in the systemWithout the retarding force, the only forces acting on the system are the gravitational forces on
and
, and the tension in the string. The net force causing the acceleration of the system is the difference in the gravitational forces on
and
, which is
.Step 2: Calculate the initial acceleration of the systemUsing Newton's second law,
, where
is the net force,
is the total mass of the system, and
is the acceleration. The total mass of the system is
. The initial acceleration
is given by:
Step 3: Introduce the retarding force
is applied to retard the upward motion of
. This force will act in the opposite direction to the acceleration, thus reducing the net force and the acceleration. The new net force is
, and the new acceleration
is:
Step 4: Determine the retarding force needed to reduce acceleration by 50%We want the new acceleration
to be half of the initial acceleration
:
Substitute the expression for
into the equation for
and solve for
:
Step 5: Solve for
Multiply both sides by \(2(m_{1} + m_{2})\) to clear the fraction:
Distribute the 2 on the right side:
Combine like terms:
Divide by -1 to get:
Subtract
from both sides:
Factor out
:
Finally, divide by 2 to solve for
:
Step 6: Plug in the values for
and
Given that
and
, we can plug these values into the equation:
**【Answer】: The value of
that will reduce the acceleration of the system by 50% is 24.50 N.**