The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket. (b) work done by gravitational force in the above case, (c) work done by friction on a body sliding down an inclined plane, (d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity, (e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

The sign of work done by a force on a body is important to understand. State carefully if the following quanti

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Q. The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.

(b) work done by gravitational force in the above case,

(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,

(e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

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Concept understanding — Dot Product Angle

The Dot Product and the Angle Between Vectors

Imagine you're pushing a heavy box across the floor. You push at an angle — not straight forward, but partly downward and partly forward. The part of your push that actually moves the box is only the forward component. The downward part just presses the box into the floor.

That's the core intuition behind the dot product: it measures how much one vector "goes in the direction of" another vector.

Step 1: What is a dot product?

Given two vectors $\vec{a}$ and $\vec{b}$ in 2D or 3D space, their dot product (also called the scalar product) is defined algebraically as:

\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3

You multiply corresponding components and add them up. The result is a single number (a scalar), not a vector.

For example, if $\vec{a} = (3, 4)$ and $\vec{b} = (2, -1)$ , then:

\vec{a} \cdot \vec{b} = 3 \times 2 + 4 \times (-1) = 6 - 4 = 2

Step 2: The geometric meaning — the angle connection

Here's the beautiful part. The dot product also has a completely different geometric definition:

\vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \, \cos \theta

where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes (lengths) of the vectors, and $\theta$ is the angle between them when they're placed tail-to-tail.

This is the dot product angle formula. It connects algebra (component multiplication) to geometry (angle and length).

Step 3: Why does this make sense?

Think about the extreme cases:

Vectors point in the same direction ( $\theta = 0^\circ$ ): $\cos 0 = 1$ , so $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|$ — the maximum possible value. All of one vector's "push" is in the other's direction.
Vectors are perpendicular ( $\theta = 90^\circ$ ): $\cos 90^\circ = 0$ , so $\vec{a} \cdot \vec{b} = 0$ . Neither vector has any component along the other. This is a crucial test for orthogonality.
Vectors point opposite ( $\theta = 180^\circ$ ): $\cos 180^\circ = -1$ , so $\vec{a} \cdot \vec{b} = -|\vec{a}| |\vec{b}|$ — the most negative value. They're completely against each other.
Any other angle: the dot product is somewhere between these extremes, proportional to how much one vector "projects" onto the other.

Tip

The dot product is positive when the angle is acute ( $< 90^\circ$ ), zero when perpendicular, and negative when obtuse ( $> 90^\circ$ ). This sign alone tells you whether the vectors are generally aligned or opposed.

Step 4: Finding the angle from the dot product

If you know the components of two vectors, you can find the angle between them by rearranging the formula:

\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \, |\vec{b}|}

Then use $\theta = \cos^{-1}(\text{that value})$ .

Example: Find the angle between $\vec{a} = (1, 2)$ and $\vec{b} = (3, 4)$ .

Compute dot product: $1 \times 3 + 2 \times 4 = 3 + 8 = 11$
Compute magnitudes: $|\vec{a}| = \sqrt{1^2 + 2^2} = \sqrt{5}$ , $|\vec{b}| = \sqrt{3^2 + 4^2} = 5$
$\cos \theta = \frac{11}{\sqrt{5} \times 5} = \frac{11}{5\sqrt{5}} \approx 0.9839$
$\theta = \cos^{-1}(0.9839) \approx 10.3^\circ$

The vectors are nearly aligned.

Watch out

The dot product formula gives $\cos \theta$ , not $\theta$ itself. Always take the inverse cosine. Also, the formula works for vectors of any dimension — 2D, 3D, even 100D — as long as you use the component definition.

Step 5: The precise statement

Important

For any two non-zero vectors $\vec{a}$ and $\vec{b}$ in $\mathbb{R}^n$ , the dot product satisfies:

$\vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \, \cos \theta$

where $\theta$ is the angle between them, $0^\circ \leq \theta \leq 180^\circ$ . This is equivalent to the component-wise definition:

$\vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i$

The two definitions are not separate — they are the same quantity expressed in different forms. The component form is how you compute it; the geometric form is what it means.

Why this matters

The dot product angle formula is the foundation for:

Projections: finding how much of one vector lies along another
Work in physics: $W = \vec{F} \cdot \vec{d}$ — only the force component along displacement does work
Orthogonality tests: two vectors are perpendicular iff their dot product is zero
Cosine similarity: used in machine learning to compare directions of data vectors

Every time you see a dot product, think: "This is telling me how aligned these two directions are."

For quick revision, remember that Dot Product Angle is a core, NCERT-aligned topic from the Work, Energy and Power (vectors) portion of the Class 11 Physics curriculum, and it is tested regularly in CBSE board exams as well as in JEE Main and NEET, which is exactly why "Dot Product Angle important questions" shows up so often in Physics question banks. The clearest way to build exam confidence here is to combine this explanation with the NCERT Physics textbook's own solved examples and chapter-end questions.

The sign of work done depends on the angle between the force and the displacement: positive if the angle is acute, negative if obtuse, and zero if perpendicular.

Positive.
Negative.
Negative.
Positive.
Negative.

Understanding the sign of work done is fundamental in physics, as it tells us whether a force is adding energy to a system, removing energy, or doing neither. Work, in physics, is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement. Mathematically, for a constant force $\vec{F}$ causing a displacement $\vec{d}$ , the work done $W$ is given by the dot product:

W = \vec{F} \cdot \vec{d} = Fd \cos\theta

Here, $F$ is the magnitude of the force, $d$ is the magnitude of the displacement, and $\theta$ is the angle between the force vector $\vec{F}$ and the displacement vector $\vec{d}$ .

The sign of $\cos\theta$ directly determines the sign of the work done:

If $0^\circ \le \theta < 90^\circ$ , then $\cos\theta$ is positive, and thus $W$ is positive. This means the force has a component in the direction of motion, aiding the motion or increasing the body's energy.
If $\theta = 90^\circ$ , then $\cos\theta = 0$ , and thus $W$ is zero. The force is perpendicular to the displacement and does no work.
If $90^\circ < \theta \le 180^\circ$ , then $\cos\theta$ is negative, and thus $W$ is negative. This means the force has a component opposite to the direction of motion, opposing the motion or decreasing the body's energy.

Let's apply this understanding to each scenario:

(a) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
- Force: The man applies an upward force on the rope, which in turn pulls the bucket upward.
- Displacement: The bucket moves upward as it is lifted out of the well.
- Angle $\theta$ : Both the applied force and the displacement are in the same upward direction. Therefore, $\theta = 0^\circ$ .
- Sign of Work: Since $\cos(0^\circ) = 1$ (positive), the work done by the man is positive. The man is expending energy to lift the bucket.
(b) Work done by gravitational force in the above case,
- Force: The gravitational force (weight) on the bucket acts vertically downward.
- Displacement: As before, the bucket moves upward.
- Angle $\theta$ : The gravitational force is downward, while the displacement is upward. They are in opposite directions. Therefore, $\theta = 180^\circ$ .
- Sign of Work: Since $\cos(180^\circ) = -1$ (negative), the work done by the gravitational force is negative. Gravity is opposing the upward motion, and the bucket is gaining gravitational potential energy.
(c) Work done by friction on a body sliding down an inclined plane,
- Force: The frictional force always opposes the relative motion between surfaces. If the body is sliding down the inclined plane, the frictional force acts up the inclined plane.
- Displacement: The body moves down the inclined plane.
- Angle $\theta$ : The frictional force is up the incline, and the displacement is down the incline. They are in opposite directions. Therefore, $\theta = 180^\circ$ .
- Sign of Work: Since $\cos(180^\circ) = -1$ (negative), the work done by friction is negative. Friction always dissipates mechanical energy as heat.
(d) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
- Force: An external force is applied to the body. Since the body moves with uniform velocity, its acceleration is zero, meaning the net force on it is zero. On a rough horizontal plane, there is a frictional force opposing motion. For uniform velocity, the applied force must be equal in magnitude and opposite in direction to the frictional force. Thus, the applied force acts in the direction of motion.
- Displacement: The body moves in the direction of its velocity.
- Angle $\theta$ : The applied force is in the direction of motion, and the displacement is also in the direction of motion. Therefore, $\theta = 0^\circ$ .
- Sign of Work: Since $\cos(0^\circ) = 1$ (positive), the work done by the applied force is positive. The applied force is doing work to overcome friction and maintain the body's motion.
Note

While the work done by the applied force is positive, the work done by friction in this case would be negative (as it opposes motion). The net work done on the body is zero, consistent with the Work-Energy Theorem, as the kinetic energy of the body remains constant (uniform velocity).
(e) Work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
- Force: The resistive force of air (air drag) always opposes the direction of motion (velocity) of the pendulum bob.
- Displacement: At any instant, the pendulum bob moves in the direction of its instantaneous velocity.
- Angle $\theta$ : The resistive force of air is always opposite to the direction of the pendulum's instantaneous displacement. Therefore, $\theta = 180^\circ$ .
- Sign of Work: Since $\cos(180^\circ) = -1$ (negative), the work done by the resistive force of air is negative. This negative work continuously removes mechanical energy from the pendulum, causing its amplitude to decrease until it eventually comes to rest.

✓Final answer

The signs of the work done are: Positive . Negative . Negative . Positive . Negative .

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Q.The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: