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Chapter 8: Problem 4

If we shift a body in equilibrium from \(A\) to \(C\) in a gravitational field via path \(A C\) or \(A B C\), (1) The work done by the force \(\vec{F}\) for both paths will be same (2) \(W_{A C}>W_{A B C}\) (3) \(W_{A C}

Short Answer

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Option (1): The work done by the force for both paths is the same.

Step by step solution

01

Understanding the Problem

We have a body in a gravitational field that needs to be moved from point \(A\) to point \(C\). There are two paths: direct path \(AC\) and path via \(B\), \(A \to B \to C\). We need to determine how the work done by the gravitational force compares between these paths.
02

Recall Work Done in Gravitational Field

The work done by the gravitational force when moving an object in a conservative gravitational field depends only on the initial and final positions, not on the path taken. This is a property of conservative forces.
03

Analyze the Paths

For both paths, the initial position is \(A\) and the final position is \(C\). Since the gravitational force is conservative, the work done, \(W_{A C}\), must be equal to the work done via \(A \to B \to C\), which means \(W_{A C} = W_{A B C}\).
04

Conclusion on Work Done

Since the work done is the same for both paths due to the conservative nature of gravitational forces, the correct choice is option (1), which states that the work done by the force \(\vec{F}\) for both paths will be the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservative Forces
In the context of physics, **conservative forces** are forces for which the work done is independent of the path taken, depending only on the starting and ending points. Gravitational force is a classic example of such a force. When dealing with conservative forces, there is a potential energy function associated with the position of the object in the field.
  • One crucial feature of conservative forces is energy conservation. The total mechanical energy (kinetic + potential) of a system under conservative forces remains constant.
  • Examples apart from gravitational force include electrostatic forces and spring forces.
This concept becomes particularly useful when assessing situations where objects move between two points in a force field. Regardless of the path taken, the work done by a conservative force like gravity is the same.
Equilibrium in Physics
A state of **equilibrium** in physics is encountered when an object is at rest or moving with constant velocity. This occurs when the net force鈥攁nd thus the net work done鈥攊s zero.
  • In static equilibrium, an object remains at rest since all acting forces balance out.
  • In dynamic equilibrium, it moves at a constant speed when forces counterbalance each other.
An object in equilibrium in a gravitational field, such as on a flat surface, experiences no net force acting to change its state of motion. When evaluating work done against gravity, subtle shifts in equilibrium are critical. Shifting an object from a point A to point C, though may move through differing pathways, those pathways don't affect the work done if an object is still within those parameters of equilibrium.
Gravitational Force
The **gravitational force** is a fundamental force of nature that attracts two bodies towards each other. This pull depends on the masses of the objects and the distance between them, described by Newton's law of universal gravitation. The formula for gravitational force is given by: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]where:
  • \(F\) is the force between the masses,
  • \(G\) is the gravitational constant,
  • \(m_1\) and \(m_2\) are the masses,
  • \(r\) is the distance between their centers.
Gravitational force is a quintessential example of a conservative force and highlights how the potential energy is path-independent when moving from one point to another. This universally attractive force provides us with the straightforward understanding that in a uniform gravitational field, the work done travelling from point A to C remains the same, irrespective of the path chosen, as long as the start and ending positions are consistent.

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Most popular questions from this chapter

The block of mass \(m\) initially at \(x=0\) is acted upon by a horizontal force \(F=a-b x^{2}\) (where \(a>\mu m g\) ), as shown in the figure. The co-efficient of friction between the surfaces of contact is \(\mu\). The net work done on the block is zero, if the block travels a distance of (1) \(x=\sqrt{\frac{3(a-\mu m g)}{b}}\) (2) \(x=\sqrt{\frac{3(a+\mu m g)}{b}}\) (3) \(x=\sqrt{\frac{2(a-\mu m g)}{b}}\) (4) \(x=\sqrt{\frac{2(a+\mu m g)}{b}}\)

A bus can be stopped by applying a retarding force \(F\) when it is moving with speed \(v\) on a level road. The distance covered by it before coming to rest is \(s\). If the load of the bus increases by \(50 \%\) because of passengers, for the same speed and same retarding force, the distance covered by the bus to come to rest shall be (1) \(1.55\) (2) \(2 s\) (3) \(1 s\) (4) \(2.5 \mathrm{~s}\)

A \(1.5-\mathrm{kg}\) block is initially at rest on a horizontal frictionless surface when a horizontal force in the positive direction of \(x\)-axis is applied to the block. The force is given by \(\vec{F}=\left(4-x^{2}\right) \vec{i} \mathrm{~N}\), where \(x\) is in meter and the initial position of the block is \(x=0\). The maximum positive displacement \(x\) is (1) \(2 \sqrt{3} \mathrm{~m}\) (2) \(2 \mathrm{~m}\) (3) \(4 \mathrm{~m}\) (4) \(\sqrt{2} \mathrm{~m}\)

A block of mass \(M\) rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass \(m\). The upper block is pushed down by an additional force \(3 \mathrm{mg}\), so the spring compression is \(4 \mathrm{mg} / \mathrm{k}\). In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. What is the greatest possible value for \(\frac{M}{m}\) ?

The potential energy of a particle is determined by the expression \(U=\alpha\left(x^{2}+y^{2}\right)\), where \(\alpha\) is a positive constant. The particle begins to move from a point with coordinates \((3,3)\), only under the action of potential field force. Then its kinetic energy \(T\) at the instant when the particle is at a point with the coordinates \((1,1)\) is (1) \(8 \alpha\) (2) \(24 \alpha\) (3) \(16 \alpha\) (4) Zero
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