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 $0.47\mathrm{m}/\mathrm{s}$. a) Calculate the net force acting on the chain. b) At the instant when $0.15\mathrm{m}$ of the chain has been lifted off the table, how much force must be applied to the end being raised?

Short answer

Answer: The force applied to lift the chain at the specific instant is approximately 1.94 N.

Step-by-step solution

Write down the given information

We are given the following values: - mass per meter of the chain: $1.32\mathrm{kg}/\mathrm{m}$ - speed with which the chain is being pulled upward: $0.47\mathrm{m}/\mathrm{s}$ - length of chain lifted off the table at a specific instant: $0.15\mathrm{m}$

Calculate the net force acting on the chain

To find the net force acting on the chain, we'll use the equation: $F_{net}=m_{lifted}\cdot a$ Here, $m_{lifted}$ is the mass of the chain that has been lifted off the table, and $a$ is the acceleration due to gravity (which is $9.81\mathrm{m}/{\mathrm{s}}^2$). To find $m_{lifted}$, we multiply the mass per meter of the chain by the length of the chain lifted off the table: $m_{lifted}=(1.32\mathrm{kg}/\mathrm{m})\cdot (0.15\mathrm{m})=0.198\mathrm{kg}$ Now, we can calculate the net force: $F_{net}=0.198\mathrm{kg}\cdot 9.81\mathrm{m}/{\mathrm{s}}^2=1.94\mathrm{N}$ So the net force acting on the chain is approximately $1.94\mathrm{N}$.

Calculate the force applied to lift the chain at the specific instant

Since the chain is being pulled upward at a constant rate, the force required to lift the chain is equal to the net force acting on the chain. Thus, at the instant when $0.15\mathrm{m}$ of the chain has been lifted off the table, the force applied to lift the chain is also approximately $1.94\mathrm{N}$.

Key concepts

Net Force

When we talk about net force, we're actually talking about the sum of all the forces acting on an object. It's like a tug-of-war where the net force is the final pull that actually makes the object move. In our chain exercise, the net force results from gravity pulling down on the chain.
This force is calculated using a simple formula:

• Net Force, $F_{net}=m_{lifted}\cdot a$

where:

• $m_{lifted}$ is the mass of the chain that has been lifted
• $a$ is the acceleration due to gravity, which is approximately $9.81\mathrm{m}/{\mathrm{s}}^2$.

Calculating net force helps us understand how much power is needed to lift parts of the chain. It considers both the mass of the portion being lifted and how quickly the force of gravity will try to bring it back down.

Uniform Chain

A uniform chain is a special type of chain where the mass is distributed evenly along its entire length. This means every portion of the chain weighs the same per meter, making it easier to calculate things like force and mass.
In the given problem, the chain has a uniform mass of $1.32\mathrm{kg}/\mathrm{m}$, meaning each meter of this chain will always weigh $1.32\mathrm{kg}$.
This uniformity makes calculations straightforward, as there is no need to calculate the mass separately for different segments of the chain. Thus, it simplifies tasks such as calculating the force needed to lift the chain or determining how much of it has lifted.

Mass and Length

Understanding mass and length in physical exercises is crucial because they directly influence force-related calculations. In our chain problem, mass is given per meter length, and that’s called the line density of the chain.
Given:

• Mass per meter = $1.32\mathrm{kg}/\mathrm{m}$
• Length lifted = $0.15\mathrm{m}$

To find the mass of the lifted chain, multiply the mass per meter by the total length lifted:$$m_{lifted}=(1.32\mathrm{kg}/\mathrm{m})\times (0.15\mathrm{m})=0.198\mathrm{kg}$$
This approach makes it easy to handle various physical concepts as real problems often involve calculating how much of a material contributes to the total mass.

Force Calculation

Understanding how to calculate force is essential for many physics problems. Force calculation often involves understanding how much effort is needed to move or lift an object. For our problem, we need to calculate how much force is necessary to lift a portion of the chain.
The required force when the chain is lifted at a constant rate is the same as the net force. This is because the forces balance each other out in steady movement, hence:

• Force applied = Net force
• $0.198\mathrm{kg}\cdot 9.81\mathrm{m}/{\mathrm{s}}^2=1.94\mathrm{N}$

So, when you pull the chain steadily upward, the force required equals the gravitational force acting on the lifted chain segment, calculated at approximately $1.94\mathrm{N}$. This understanding of force makes it possible to figure out how to move or lift objects efficiently.