Three sleds are being pulled horizontally on frictionless horizontal ice using horizontal ropes (Fig. E$5.14$). The pull is of magnitude $190$ N. Find the tension in ropes $A$ and $B$.

Verified step by step guidance

1. Step 1: Identify the forces acting on the sleds. The pull force of 190 N acts horizontally to the right, and the tension in ropes A and B transmits this force between the sleds. Since the ice is frictionless, there is no opposing force due to friction.
2. Step 2: Calculate the total mass of the system. Add the masses of all three sleds: \( m_{total} = m_1 + m_2 + m_3 \). Using MathML: $m$_total=30+20+10
3. Step 3: Determine the acceleration of the system. Use Newton's second law \( F = ma \) to find the acceleration \( a \). Rearrange to \( a = F / m_{total} \). Using MathML: $a=\frac{F}{m}$_total
4. Step 4: Calculate the tension in rope A. Rope A is responsible for pulling the second and third sleds (20 kg + 10 kg). Use Newton's second law \( T_A = (m_2 + m_3) \cdot a \). Using MathML: $T$_A=(m₂+m₃)⋅a
5. Step 5: Calculate the tension in rope B. Rope B is responsible for pulling only the third sled (30 kg). Use Newton's second law \( T_B = m_3 \cdot a \). Using MathML: $T$_B=m₃⋅a

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, understanding how the total force of 190 N affects the sleds' acceleration is crucial for calculating the tensions in the ropes.

Tension in Ropes

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In this problem, the tension in ropes A and B must be calculated based on the forces acting on the sleds they connect. The tension will vary depending on the mass of the sleds and the acceleration they experience due to the applied force.

System of Connected Objects

When analyzing a system of connected objects, such as the sleds in this problem, it is essential to consider the entire system's mass and the forces acting on each individual object. By applying Newton's laws to the system as a whole and to each sled separately, one can derive equations that relate the tensions in the ropes to the total force and the masses of the sleds, allowing for the calculation of unknown tensions.