Understanding the Problem
In this scenario, we have two objects connected by a rope passing over a pulley.
- Block 1 ($M = 4 \text{ kg}$): Sits on a horizontal, frictionless surface.
- Block 2 ($m$ = ?): Suspended vertically by the rope.
- Tension ($T = 10 \text{ N}$): The rope is light (massless) and the pulley is frictionless.
Since both blocks are connected by the same rope, they will accelerate together with the same magnitude of acceleration, denoted as $a$.
Free Body Diagrams (FBD)
Block 1 (Horizontal Mass)
- Vertical forces: Gravity ($Mg$) acts downwards, and Normal force ($N$) acts upwards. Since there is no vertical motion, $N = Mg$.
- Horizontal force: Only tension ($T$) acts to pull the block forward.
- Equation: $T = Ma$
Block 2 (Hanging Mass)
- Vertical forces: Gravity ($mg$) acts downwards, and Tension ($T$) acts upwards.
- Since the mass is hanging and the system accelerates downward:
- Equation: $mg - T = ma$
Step-by-Step Solution
1. Calculate Acceleration ($a$)
Using the horizontal block's equation: $T = Ma$ $10 \text{ N} = (4 \text{ kg}) \cdot a$ $a = \frac{10}{4} = 2.5 \text{ m/s}^2$
2. Calculate Hanging Mass ($m$)
Using the hanging block's equation: $mg - T = ma$ $m(g - a) = T$ Taking $g \approx 9.8 \text{ m/s}^2$: $m(9.8 - 2.5) = 10$ $m(7.3) = 10$ $m = \frac{10}{7.3} \approx 1.37 \text{ kg}$
Intuition
The tension in the rope must be lower than the weight of the hanging block ($mg$) because the hanging block is accelerating downwards. If the system were static, tension would equal weight. Because it accelerates, the "net force" is $mg - T$, which results in motion.