Question

A uniform chain with a mass of \(1.32 \mathrm{~kg}\) per meter of length is coiled on a table. One end is pulled upward at a constant rate of \(0.470 \mathrm{~m} / \mathrm{s}\). a) Calculate the net force acting on the chain. b) At the instant when \(0.150 \mathrm{~m}\) of the chain has been lifted off the table, how much force must be applied to the end being raised?

Short answer

The net force acting on the chain is 1.32 L × 9.81 N, and the force that must be applied to the end being raised when 0.15 meters of the chain is lifted off the table is approximately 1.94 N.

Step-by-step solution

Calculate the mass of the chain being lifted off the table

The chain is being lifted off the table at a constant speed of \(0.47\,\mathrm{m/s}\). We are given the mass per length of the chain as \(1.32\,\mathrm{kg/m}\). We'll use this to calculate the mass being lifted at any given instant. Let's represent the length of the chain being lifted with variable \(L\). The mass \(m\) of the chain being lifted is given by: $$ m = 1.32 L $$

Calculate the weight of the lifted portion of the chain

The weight of the lifted portion of the chain can be obtained by multiplying the mass with the acceleration due to gravity \(g \approx 9.81\,\mathrm{m/s^{2}}\). So, the weight of the lifted portion of the chain is: $$ W = m \cdot g = 1.32 L \cdot 9.81 $$

Calculate the net force acting on the chain

Since the chain is being lifted at a constant speed, the net force acting on the chain is equal to the weight of the lifted portion of the chain. So, the net force acting on the chain \(F_{net}\) is: $$ F_{net} = W = 1.32 L \cdot 9.81 $$

Calculate the force applied to the end being raised

We need to find the force \(F\) applied to the end being raised when \(0.150\,\mathrm{m}\) of the chain has been lifted off the table. At this instant, the length of the chain being lifted \(L = 0.150\,\mathrm{m}\). Using this value, we can calculate the force applied to the end being raised: $$ F = F_{net} = 1.32 \times 0.150 \times 9.81 \approx 1.94\,\mathrm{N} $$ a) The net force acting on the chain is \(1.32 L \cdot 9.81\,\mathrm{N}\). b) The force that must be applied to the end being raised when \(0.15\,\mathrm{m}\) of the chain is lifted off the table is approximately \(1.94\,\mathrm{N}\).

Key concepts

Uniform Chain

The term 'uniform chain' refers to a chain that has a consistent mass per unit length throughout its entire length. This is an important aspect when addressing physics problems because it allows for a simplification of calculations. In our scenario, the uniform chain helps us calculate the mass of any segment of the chain easily, by knowing its length and mass per meter. This is particularly useful when we are dealing with a scenario of lifting a portion of such a chain at a constant rate, as the mass is directly proportional to the length of the chain lifted from the surface.

Understanding the properties of a uniform chain is vital for accurately determining the forces involved when interacting with the chain. In real-world applications, this might translate to scenarios like material handling in industrial settings or understanding the mechanics of jewelry design, where uniformity is a desired trait for both functional and aesthetic reasons.

Constant Speed Lifting

When an object is lifted at a constant speed, it indicates that the object's velocity is uniform and does not change over time. For a chain being lifted at a constant speed, it implies that there is a balance between the forces acting upon it—specifically, the force of the lift and the force of gravity. The net force, in this case, is zero, which means the acceleration of the chain is also zero.

Lifting at a constant speed is essential in many industrial processes, where precision and control are imperative. For students trying to understand this concept, it's akin to pulling an object with a consistent effort without speeding up or slowing down, hence not allowing the object to gain or lose speed.

Weight of Lifted Portion

When calculating the weight of the lifted portion of a chain, we are essentially focusing on the segment of the chain that has been removed from a resting position and is now under the influence of gravity alone. The weight of this portion is determined by the product of its mass and the acceleration due to gravity. In physical terms, it's the downward force that needs to be counteracted in order to lift the chain at a constant speed.

To visualize this, think of the lifted portion of the chain as an independently hanging object whose weight is solely due to its interaction with Earth's gravity. The force exerted by a person or machine to counteract this weight must match it exactly to maintain constant speed lifting, highlighting the practical significance of calculating weight in mechanics and engineering applications.

Acceleration Due to Gravity

Acceleration due to gravity is a constant value that represents the acceleration that any object experiences when in free fall towards the Earth's surface. It is approximately measured as \(9.81 \text{ m/s}^2\) and is symbolized by the letter \(g\). This value plays a critical role in numerous physical calculations because it determines the force of gravity acting on objects.

All objects experience this acceleration irrespective of their mass, assuming negligible air resistance. For students, understanding this concept is crucial as it forms the basis for explaining why objects fall to the ground and how force must be applied to counteract this natural phenomenon. Whether designing a roller coaster or understanding the motion of planets, gravity's acceleration is a fundamental concept engrained in the fabric of physics.