Work Energy And Power Class 11 One Shot | Detailed Revision | Physics Chapter 5 | By Akshay Sir

Summary

The lecture reviews the main ideas in Work, Energy and Power, then applies them to vertical circular motion and collisions. The central theme is that work describes energy transfer, while energy methods often let you solve motion problems without finding every force and acceleration separately.

1. Work and its direction

• In physics, work is not simply “effort” or distance travelled. A force does work when it acts while an object is displaced.

• Work is the dot product of force and displacement: [ W=\vec F\cdot\vec s=Fs\cos\theta ] where (\theta) is the angle between the force and displacement. Work is a scalar.

• If several forces act, calculate the work done by each force separately:

  • A force in the direction of displacement does positive work.
  • A force opposite the displacement does negative work. Friction often does negative work.
  • A force perpendicular to the displacement does zero work. For example, the normal force does no work on a box sliding along a horizontal surface; gravity also does no work in that situation.
• For an object moving vertically, gravity does (+mgh) of work when the object moves down by height (h), and (-mgh) when it moves up by that height.
• The SI unit of work is the joule, with dimensions (ML^2T^{-2}).

2. Work by a variable force

When force changes with position, use integration rather than (Fs):

[ W=\int_{x_i}^{x_f}F(x)\,dx ]

The work is also the area under the force–displacement graph. For example, because spring force changes with extension, spring work must be treated as variable-force work.

3. Kinetic energy, potential energy, and the work–energy theorem

• Kinetic energy is the energy an object has because of its motion: [ K=\frac12mv^2 ] Using momentum (p=mv), this can also be written: [ K=\frac{p^2}{2m} ]

• Potential energy depends on the system’s position and the chosen reference level. For gravity near Earth: [ U=mgh ] The zero level is arbitrary; what matters physically is the change in potential energy, (\Delta U).

• The work–energy theorem states: [ W_{\text{all forces}}=\Delta K ] In other words, the total work done by all forces equals the change in kinetic energy.

• For a conservative force: [ W_{\text{conservative}}=-\Delta U ] Gravity and ideal spring force are examples of conservative forces. Their work depends only on the initial and final positions, not the route taken.

• Friction is non-conservative: its work depends on the path. In the carrom-board example, friction does negative work along each segment, so the total work is not zero even if the object returns to its starting point.

• If only conservative forces do work, mechanical energy is conserved: [ K_i+U_i=K_f+U_f ] This is useful for problems involving falling objects, springs, or changes in height.

4. Spring potential energy

For a spring with spring constant (k), Hooke’s law gives the restoring force (F=-kx). The spring’s potential energy at extension or compression (x) is:

[ U_s=\frac12kx^2 ]

The lecture derives this by integrating the variable spring force. Because energy is proportional to (x^2), stretching the same spring from (2\text{ cm}) to (8\text{ cm}) increases its stored energy by a factor of 16.

5. Force and potential-energy graphs

• In one dimension, force is related to potential energy by: [ F=-\frac{dU}{dx} ] The negative sign means the force points toward decreasing potential energy.

• At a stationary point of a potential-energy graph, (dU/dx=0). The second derivative helps classify it:

  • (d^2U/dx^2>0): local minimum, generally a stable equilibrium.
  • (d^2U/dx^2<0): local maximum, generally an unstable equilibrium.
• The instructor demonstrates finding a force from a given (U(x)) and locating where potential energy is minimum by differentiating.

6. Power and energy units

• Power is the rate at which work is done: [ P_{\text{avg}}=\frac{W}{t},\qquad P_{\text{instantaneous}}=\frac{dW}{dt}=\vec F\cdot\vec v ] Power is a scalar; its SI unit is the watt.

• Since (W=Fs), dividing by time gives the force–velocity relation for power.

• The commercial unit of energy is the kilowatt-hour: [ 1\text{ kWh}=3.6\times10^6\text{ J} ] The lecture also notes that (1) horsepower is approximately (746\text{ W}).

• In a pump example, the method is to equate power to the rate of increase of gravitational potential energy: [ P=\frac{mgh}{t} ] Efficiency is output power divided by input power.

7. Vertical circular motion

For a mass moving on a vertical circle of radius (r), the instructor writes the radial force equation, with (\theta) measured from the bottom of the circle:

[ T-mg\cos\theta=\frac{mv^2}{r} ]

Thus, tension varies with both speed and position. At the top, (\theta=180^\circ), so:

[ T=\frac{mv_{\text{top}}^2}{r}-mg ]

For the mass to just complete the circle, the limiting condition at the top is (T=0). Therefore:

[ v_{\text{top}}=\sqrt{gr} ]

Applying conservation of mechanical energy between the bottom and top gives the minimum speed at the bottom:

[ v_{\text{bottom}}=\sqrt{5gr} ]

The lecture also uses energy conservation for a mass that rises only to a particular height and momentarily stops. The general approach is to compare kinetic plus potential energy at the starting point and at the endpoint.

8. Collisions

• Momentum is conserved in an isolated collision: [ \sum \vec p_{\text{before}}=\sum \vec p_{\text{after}} ]

• Elastic collision: momentum and kinetic energy are both conserved.

• Inelastic collision: momentum is conserved, but kinetic energy is not.

• Perfectly inelastic collision: the objects stick together and share a common final velocity. In one dimension: [ m_1u_1+m_2u_2=(m_1+m_2)v ]

• The coefficient of restitution is the ratio of relative speed of separation to relative speed of approach. In one dimension: [ e=\frac{\text{speed of separation}}{\text{speed of approach}} ] (e=1) corresponds to an elastic collision, (e=0) to a perfectly inelastic collision, and intermediate values to partially inelastic collisions.

• For a two-dimensional collision, conserve momentum separately along the (x)- and (y)-axes. Apply kinetic-energy conservation as well if the collision is elastic.

• In an elastic one-dimensional collision between equal masses, the velocities are exchanged.

9. Problem-solving strategies and examples

The instructor emphasizes choosing the method that fits the information given:

• Use (W=Fs\cos\theta) for a constant force.
• Use (\int F(x)\,dx) for a variable force.
• Use (W_{\text{all}}=\Delta K) when work or a force is given and the question asks about speed or kinetic energy.
• Use (K+U=\text{constant}) when only conservative forces act and the problem involves changing height, speed, or spring compression.
• Use momentum conservation for collisions; add kinetic-energy conservation only for elastic collisions.
• Check direction and signs carefully, especially for gravity, friction, and forces opposite the displacement.

Examples in the lecture include work by an angled force, integration of a position-dependent force, spring compression, an object’s energy as it falls, projectile kinetic energy at its highest point, and momentum and kinetic-energy comparisons.

10. Study advice

The instructor asks students to review five chapters over October and November, solve roughly 50 questions per chapter, and include theory and derivations rather than practising numericals alone. He encourages students not to give up if concepts take more than one viewing.

Speakers and sources featured

• One speaker: The physics instructor, who introduces himself in the subtitles as “Akshay Dayal” and says students can call him “ATB Bhaiya.” The video metadata identifies the lecturer as Akshay Tyagi/Akshay Sir; the subtitles appear to render his spoken name differently.
• No other speakers or named external sources are featured.