Understanding Work Done Against Gravity

The question asks about the work done against gravity when a body moves horizontally along a frictionless surface. To solve this, we need to understand the definition of work done in physics.

Definition of Work Done

Work done by a force on an object is defined as the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and the displacement vectors.

Mathematically, work done ($W$) is given by:

\(W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos(\theta)\)

Where:

• \(\vec{F}\) is the force vector
• \(\vec{d}\) is the displacement vector
• \(|\vec{F}|\) is the magnitude of the force
• \(|\vec{d}|\) is the magnitude of the displacement
• \(\theta\) is the angle between the force vector and the displacement vector

Analyzing the Scenario

In this specific scenario, we are considering the work done against gravity. This means the force we are interested in is the force of gravity acting on the body.

• The force of gravity always acts vertically downwards towards the center of the Earth.
• The body is moved horizontally along a surface. This means the displacement vector is horizontal.

Determining the Angle Between Force and Displacement

The force of gravity is vertical, and the displacement is horizontal. A vertical direction is always perpendicular to a horizontal direction. Therefore, the angle (\(\theta\)) between the force of gravity (\(\vec{F}_g\)) and the horizontal displacement (\(\vec{d}\)) is 90 degrees.

\(\theta = 90^\circ\)

Calculating the Work Done Against Gravity

Now, we can use the formula for work done:

\(W = |\vec{F}_g| |\vec{d}| \cos(\theta)\)

Substitute the angle \(\theta = 90^\circ\):

\(W = |\vec{F}_g| |\vec{d}| \cos(90^\circ)\)

We know that the cosine of 90 degrees is 0:

\(\cos(90^\circ) = 0\)

So, the work done against gravity is:

\(W = |\vec{F}_g| |\vec{d}| \times 0\)

\(W = 0\)

Regardless of the magnitude of the force of gravity (\(|\vec{F}_g|\)) or the horizontal displacement (\(|\vec{d}|\)), if the angle between them is 90 degrees, the work done by or against that force is zero.

The fact that the surface is frictionless is not relevant to the work done specifically against gravity. Friction would affect the force required to move the body horizontally, but not the work done against the vertical force of gravity during horizontal motion.

Conclusion

When a body is moved horizontally, the force of gravity acting on it is perpendicular to the direction of motion. Because the angle between the force and displacement is 90 degrees, the work done against gravity is zero.