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A source of energy is one which is capable of providing an adequate amount of useful energy.

The gravitational potential energy of an object at a point above the ground is defined as the work done in raising the object from the ground to that point against gravity. G.P.E. = mgh.

The unit of power is watt. One watt is defined as the power when one joule of work is done in one second.

Work: Work is said to be done by the force when the force applied on a body displaces it. 

Energy: Energy is defined as the ability to do work. 

Power: The rate of work done is called power.

(d) all of the above.

The average power is defined as the ratio of the total work done to the total time taken. 

P_av = total work done/total time taken The instantaneous power is defined as the power delivered at an instant

p_inst = dw/dt

When two colliding bodies are of the same mass, there will be maximum exchange of energy.

Elastic potential Energy,

\(U=\frac{1}{2}kx^2\)

where k is called spring constant.

If roads go straight up then angle of slope 0 would be large so frictional force f = µ mg cos θ would be less and the vehicles may slip. Also greater power would be required.

The total mechanical energy of a system remains same, if the working force on it are of conservative nature.

The sum of kinetic energy and potential energy of a system is called mechanical energy. For a given system, the sum of kinetic energy and potential energy remains constant. When kinetic energy increases then potential energy decreases and vice versa is also true. This explains the conservation of mechanical energy of a given system. This can be understood by example of a stone kept at a height. The potential energy of stone is maximum and kinetic energy is zero at that height. When the stone is in free fall, its kinetic energy increases and potential energy decreases. When the stone hits the ground, its kinetic energy becomes maximum and potential energy becomes zero. Throughout this, the mechanical energy remains constant.

If roads were to go straight up, the slope (θ) would have been large, the frictional force (μmg cos θ) would be small. The wheels of the vehicle would slip. Also for going up a large slope a greater power shall be required.

There are two laws:

(i) Law of conservation of energy: For an isolated system or body in presence of conservative forces the sum of KE and PE at any point remains constant throughout the motion.

(ii) Law of conservation of total energy : If the forces are conservative and non-conservative both, it is not the mechanical energy alone which is conserved, but it is the total energy, may be heat, light, sound, or mechanical etc., which is conserved.

Kinetic energy increases

Potential energy will increase. This is because in bringing two protons closer, work has to be done against the force of repulsion. This is stored up in the form of potential energy.

However, the potential energy will decrease when a proton and an electron are brought nearer. Work will be done by the force of attraction between them.

(i) When the same charged particles are brought towards each other, the potential energy of the system will increase. Because work has to be done against the force of repulsion. This work done only stored as potential energy. 

(ii) When two oppositely charged particles are brought towards each other, the potential energy of the system will decrease. Because work is done by the force of attraction between the charged particles.

Work has to be done to overcome the force of repulsion. Thus work done will stored in the form of P.W. So, it increase the potential energy of the system.

The vehicle stops when its K.E. is spent in working against the force as friction between the tyres and the road. This force of friction varies directly with the weight of the vehicle.

As, K.E. = work done

= Force of friction × distance

or E = f × s

or s = E/f

For given K.E., s, will be smaller where F is larger is such as in the case of truck.

Power is defined as the rate at which the body can do the work.

P = \(\frac{\text{work done}}{\text{Time taken}}\)

Mass of cube, m = 50g = 5.0 × 10^-2 kg

Speed of cube, v = 10 cm/s = 1.0 × 10^-1 m/s

Young’s modulus Y = 2.0 × 10¹¹ N/m²

Side of cube(L) = 1 cm = 1.0 × 10^-2 m

Apply Hooke’s Law,

Young modulus Y = \(\frac{\frac{F}{A}}{\frac{ΔL}{L}}\)

So, \(\frac{F}{ΔL} = \frac{YA}{L} ........(i)\)

Or, \(\frac{F}{ΔL} = k ........(ii)\)

From eqn (i) & eqn (ii)

\(k=\frac{YA}{L} =\frac{YL^2}{L},\)

(Here A = L²)

K = YL

Initial KE = \(2\times \frac{1}{2}mv^2=5.0\times 10^{-4}J\)

Final PE = \(2\times \frac{1}{2}k​​ΔI^2\)

= kΔI² = YLΔI²

Apply Law of conservation of energy

YLΔI² = 5.0 × 10^-4

or, ΔL = \(\sqrt \frac{5.0\times 10^{-4}}{YL}\)

= \(\sqrt \frac{5.0\times 10^{-4}}{(2.0\times 10^{11}\times 10^{-2})}m\)

Δl = 5.0 × 10^-7 m

Using F.s = \(\frac{1}{2}mv^2-\frac{1}{2}mu^2\)

where m = 10 g = \(\frac{10}{1000}\)kg

= 0.01 kg

and v = 100 ms^-1 , u = 800 ms^-1

and s = 1 m,

we get F = \(\frac{1}{2}\) × 0.01 × (100² − 800²)

= −3150 N.

For x = 10 cm = 0.1 m, Fx = \(\frac{1}{2}\)\(mv^2_1\) = 1J

∴ F = 10N

For x = 6 cm = 0.06 m, Fx = \(\frac{1}{2} mv^2_1 - \frac{1}{2}{mv^2_2}\)

or Fx = \(\frac{1}{2}\) \(mv^2_1\) - Final K.E.

Final K.E. = \(\frac{1}{2}\) \(mv^2_1\) - Fx = 1 - 10 x 0.06 =1 - 0.06 = 0.4 J

Correct answer is (c) energy

(a) Velocity of the particle

As energy associated with discharge of a single neutron is 10^-10 J, therefore total energy in a nerve impulse, where 10⁵ neurons are fired is 10^-10 × 10⁵ = 10^-5 J.

Correct answer is (c) Solar

Two conditions required to do work are force applied and displacement produced.

In slow operation, weight acts on the centre of gravity of the spring.

In free fall the same weight acts at the lower most end of the spring. The same spring can now have change in double of length, i.e., 2l.

(i), (ii), (iii), (iv), and (vi)

The potential energy of a system of two masses is inversely proportional to the separation between them. In the given case, the potential energy of the system of the two balls will decrease as they come closer to each other. It will become zero (i.e., V(r) = 0) when the two balls touch each other, i.e., at r = 2R, where R is the radius of each billiard ball. The potential energy curves given in figures (i), (ii), (iii), (iv), and (vi) do not satisfy these two conditions. Hence, they do not describe the elastic collisions between them.

In inelastic collision total energy is only conserved but kinetic energy is not conserved. A part of kinetic energy is converted into some other form of energy such as sound, heat energy. Note: The linear momentum is also conserved.

False

Although internal forces are balanced, they cause no work to be done on a body. It is the external forces that have the ability to do work. Hence, external forces are able to change the energy of a system.

True

In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system. This is because in such collisions, there is always a loss of energy in the form of heat, sound, etc.

False

The work done in the motion of a body over a closed loop is zero for a conservation force only.