Question55. (II) Figure 4-49 shows a block (mass ) on a smooth horizontal surface, connected by a thin cord that passes over a pulley to a second block ( ), which hangs vertically. (a) Draw a free-body diagram for each block, showing the force of gravity on each, the force (tension) exerted by the cord, and any normal force. (b) Apply Newton's second law to find formulas for the acceleration of the system and for the tension in the cord. Ignore friction and the masses of the pulley and cord.
FIGURE 4-49
Problems 55, 56, and 57. Mass rests on a smooth
horizontal
surface;
hangs vertically.
Studdy Solution
STEP 1
1. The surface on which block rests is smooth, implying no friction.
2. The pulley and cord are massless.
3. The only forces acting on the blocks are gravity, tension, and normal force.
4. We need to find the acceleration of the system and the tension in the cord.
STEP 2
1. Draw free-body diagrams for each block.
2. Apply Newton's second law to each block.
3. Solve the system of equations for acceleration and tension.
STEP 3
Draw free-body diagrams for each block.
- For block on the horizontal surface:
- Gravity force acting downward.
- Normal force acting upward.
- Tension force acting horizontally to the right.
- For block hanging vertically:
- Gravity force acting downward.
- Tension force acting upward.
STEP 4
Apply Newton's second law to each block.
- For block :
$ \sum F_x = m_A \cdot a = T
\]
- For block :
$ \sum F_y = m_B \cdot a = m_B \cdot g - T
\]
STEP 5
Solve the system of equations for acceleration and tension.
From block :
From block :
Substitute into the equation for block :
Combine terms:
Solve for acceleration :
Substitute back into the equation for :
The formulas for the acceleration of the system and the tension in the cord are:
- Acceleration:
- Tension:
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