Two boxes of fruit on a frictionless horizontal surface are connected by a light string as in the figures below, where m1 = 10 kg and m2 = 19 kg. A force of 46 N is applied to the 19 kg box.
Repeat the problem for the case where the coefficient of kinetic friction between each box and the surface is 0.09
To solve this problem, we need to consider the forces acting on the boxes and apply Newton's laws of motion. Let's break it down step by step:
Step 1: Draw a free-body diagram to identify the forces acting on each box.
In this case, we have two boxes connected by a string. Let's call the box with mass 10 kg as m1 and the box with mass 19 kg as m2. The force applied to the 19 kg box is 46 N.
+----m1----+
| |
T1 | | T2
| |
+----m2----+
m1:
- Force due to gravity (weight) = m1 * g (where g is the acceleration due to gravity)
- T1 (tension force acting to the left, pulling m1)
m2:
- Force due to gravity (weight) = m2 * g
- T2 (tension force acting to the right, pulling m2)
- Force due to the applied force = 46 N (to the right)
Step 2: Set up equations of motion for each box using Newton's second law (F = ma).
For m1:
Summing forces in the horizontal direction:
T1 = m1 * a1 (equation 1)
For m2:
Summing forces in the horizontal direction:
T2 - F(friction) = m2 * a2 (equation 2)
Since the boxes are connected by a string, their accelerations are the same: a1 = a2 = a.
Step 3: Determine the tension forces.
In this case, the tension is the same throughout the string.
For m1:
T1 = m1 * a (equation 3)
For m2:
T2 - F(friction) = m2 * a (equation 4)
Now let's repeat the problem considering the coefficient of kinetic friction is 0.09.
Step 1: Draw a free-body diagram to identify the forces acting on each box.
The forces acting on the boxes remain the same as in the previous case, with an additional force due to kinetic friction acting in the opposite direction to the motion.
m1:
- Force due to gravity (weight) = m1 * g
- T1 (tension force acting to the left, pulling m1)
- Force due to friction = μ * Normal force (where μ is the coefficient of kinetic friction)
m2:
- Force due to gravity (weight) = m2 * g
- T2 (tension force acting to the right, pulling m2)
- Force due to the applied force = 46 N
Step 2: Set up equations of motion for each box using Newton's second law (F = ma).
For m1:
Summing forces in the horizontal direction:
T1 - F(friction) = m1 * a1 (equation 1)
For m2:
Summing forces in the horizontal direction:
T2 - F(friction) - F(applied) = m2 * a2 (equation 2)
Since the boxes are connected by a string, their accelerations are the same: a1 = a2 = a.
Step 3: Determine the tension forces.
In this case, the tension is the same throughout the string.
For m1:
T1 - F(friction) = m1 * a (equation 3)
For m2:
T2 - F(friction) - F(applied) = m2 * a (equation 4)
By solving equations 1 and 3 simultaneously, and then equations 2 and 4 simultaneously, we can find the values of the tensions and accelerations for each case.