Question

Two blocks of equal mass are connected by a massless horizontal rope and resting on a frictionless table. When one of the blocks is pulled away by a horizontal external force \(\vec{F},\) what is the ratio of the net forces acting on the blocks? a) 1: 1 b) 1: 1.41 c) 1: 2 d) none of the above

Short answer

Answer: (a) 1:1

Step-by-step solution

Identify the forces on each block

We have two forces acting on the blocks: external force \(\vec{F}\) acting on the first block and tension force \(\vec{T}\) acting on both blocks due to the rope. Let's denote the blocks as A and B. For block A: - External force \(\vec{F}\) acting horizontally - Tension force \(\vec{T}\) acting horizontally in the opposite direction For block B: - Tension force \(\vec{T}\) acting horizontally

Apply Newton's Second Law to both blocks

Newton's second law states that the net force acting on an object is equal to the mass of that object multiplied by its acceleration: \(\vec{F}_{net} = m\vec{a}\). For block A, we'll have: \(m\vec{a}_A = \vec{F} - \vec{T}\) For block B, we'll have: \(m\vec{a}_B = \vec{T}\) Note that since the rope is massless and inextensible, both blocks will have the same magnitude of acceleration, i.e., \(|\vec{a}_A| = |\vec{a}_B|\).

Solve for the tension force and acceleration

To solve for the tension force and acceleration, first, add the two equations we got in Step 2: \(m\vec{a}_A + m\vec{a}_B = \vec{F}\) Since \(|\vec{a}_A| = |\vec{a}_B|\), we'll have: \(2m\vec{a} = \vec{F}\) Solving for acceleration \(\vec{a}\), we get: \(\vec{a} = \frac{1}{2}\frac{\vec{F}}{m}\) Now, substitute this acceleration value into the equation for block B: \(m\vec{a}_B = \vec{T}\) \(m\left(\frac{1}{2}\frac{\vec{F}}{m}\right) = \vec{T}\) \(\frac{1}{2}\vec{F} = \vec{T}\)

Calculate the ratio of net forces

Now we have the expressions for the net forces on each block. For block A, the net force is \(\vec{F}_{netA} = \vec{F} - \vec{T}\), and for block B, the net force is \(\vec{F}_{netB} = \vec{T}\). The ratio of the net forces is: \(\frac{\vec{F}_{netA}}{\vec{F}_{netB}} = \frac{\vec{F} - \vec{T}}{\vec{T}}\) Using the fact that \(\vec{T} = \frac{1}{2}\vec{F}\), we can rewrite the ratio: \(\frac{\vec{F}_{netA}}{\vec{F}_{netB}} = \frac{\vec{F} - \frac{1}{2}\vec{F}}{\frac{1}{2}\vec{F}} = \frac{\frac{1}{2}\vec{F}}{\frac{1}{2}\vec{F}}\)

Simplify the ratio expression

We can now cancel out the force term in the ratio expression: \(\frac{\vec{F}_{netA}}{\vec{F}_{netB}} = \frac{\frac{1}{2}\vec{F}}{\frac{1}{2}\vec{F}} = 1\) The ratio of the net forces acting on the blocks is 1:1. Therefore, the answer is (a) 1:1.

Key concepts

Net Force Calculation

Understanding how to calculate the net force is crucial in physics as it determines the overall effect of multiple forces acting on an object. When we talk about the net force, we mean the vector sum of all forces acting on an object. It is what causes an object to accelerate in accordance with Newton's Second Law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (\( \vec{F}_{net} = m\vec{a} \)).

Let's consider a scenario from our exercise where two blocks connected by a rope lay on a frictionless surface. If an external force is applied to one of the blocks, the net forces on both blocks are affected. To find the net force acting on each block, we sum up all the individual forces. In our case, block A has the external force and the opposite tension, while block B only experiences the tension. The key is to understand that these forces must be added vectorially, not just numerically, to account for their directions.

Tension Force in Physics

The tension force is an example of a contact force that arises in a string, rope, cable, or similar when it's pulled tight by forces acting from opposite ends. It's transmitted through the string and acts along its length. In physics problems, we often assume the rope to be massless and inextensible, meaning it doesn't stretch. This assumption allows us to state that the tension is the same throughout the entire length of the rope, and the acceleration of any objects connected by such a rope will be identical.

In our textbook problem, when the external force pulls on block A, the tension force not only opposes the external force on block A but also acts on block B, pulling it in the direction of the applied force. The beauty of tension in physics problems is its ability to transmit force and create a relationship between the motions of different objects interconnected by the rope.

Frictionless Surface Physics

When tackling problems with frictionless surfaces, it's important to note that such surfaces are idealized scenarios. They don't exist in the real world but are used in physics to simplify problems and focus on the basics of motion and forces. On a frictionless surface, there is no resistance to the movement of objects, meaning no force opposes the motion parallel to the surface. As a result, the only forces we need to consider are those acting perpendicular to the surface (like normal force) and any applied forces (like tension or external forces).

In the context of our exercise, the assumption of a frictionless surface meant we didn't have to consider the frictional force that would typically act on the blocks, thus simplifying the force analysis and calculation of net forces. When dealing with real-world applications, always remember that friction is a force that resists motion, and its absence in these problems is only an assumption for simplicity and should not be misinterpreted as a physical reality.