JEE: Power with Variable Force 💡

 

❓ Concept Question

How do we calculate instantaneous power when the force is time-dependent?

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🖼 Concept Image

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✍️ Short Concept

Power represents rate of doing work.

For a moving particle:

$$P = \vec{F} \cdot \vec{v}$$

If force varies with time, velocity must be found using Newton’s law and integration.

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🔷 Step 1 — Instantaneous Power Definition 💯

Power is defined as:

$$P = \vec{F} \cdot \vec{v}$$

This is a dot product.

👉 Only the component of force along velocity contributes to power.

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🔷 Step 2 — When Force Depends on Time

If:

$$\vec{F} = F(t)$$

Then acceleration becomes:

$$\vec{a} = \frac{\vec{F}}{m}$$

Since velocity changes with time:

$$\vec{v} = v(t)$$

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🔷 Step 3 — Finding Velocity

Using Newton’s second law:

$$F = ma$$

and

$$a = \frac{dv}{dt}$$

So:

$$\frac{dv}{dt} = \frac{F(t)}{m}$$

Integrating:

$$v = \int \frac{F(t)}{m} \, dt$$

👉 Integration gives velocity.

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🔷 Step 4 — Apply Dot Product

If force is vector:

$$\vec{F} = F_x \hat{i} + F_y \hat{j}$$

and velocity:

$$\vec{v} = v_x \hat{i} + v_y \hat{j}$$

Then power becomes:

$$P = F_x v_x + F_y v_y$$

⚠️ Direction always matters.

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🔷 Step 5 — JEE Golden Traps

❌ Directly writing $P = Fv$

❌ Ignoring vector direction

❌ Treating time-dependent force as constant

👉 Always check dot product and velocity expression.

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✅ Final Takeaway

$$\boxed{P = \vec{F} \cdot \vec{v}}$$

For time-dependent force:

1️⃣ Find velocity using integration
2️⃣ Apply dot product

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⭐ Golden JEE Insight

If force and velocity are perpendicular:

$$P = 0$$

Example:

👉 Uniform circular motion.

Force exists but power is zero.