All questions of Work, Energy and Power for NEET Exam

Explanation:
When work is done on a system against a conservative force, the potential energy of the system increases.

- Conservative Force: A force is said to be conservative if the work done by the force on a particle moving from one point to another depends only on the initial and final positions of the particle and not on the path followed by the particle.
- Potential Energy: Potential energy is the energy possessed by a system due to the relative positions of its components. It is a scalar quantity and is measured in joules (J).

In the given options, only option D mentions work done by the system against a conservative force, which is the correct answer.

When a system does work against a conservative force, the energy is stored in the system as potential energy. This potential energy can be released later when the system returns to its original position or configuration.

For example, when a spring is compressed by an external force, the system stores potential energy. When the external force is removed, the spring returns to its original position and releases the stored potential energy.

Hence, the potential energy of a system increases when work is done by the system against a conservative force.

**Answer:**

The power exerted by the gravitational force can be calculated using the formula:

Power = force x velocity

When a body is projected vertically upwards, the only force acting on it is the gravitational force. The gravitational force is given by:

F = mg

Where:
F = gravitational force
m = mass of the body
g = acceleration due to gravity

Since the body reaches a height equal to the Earth's radius, the distance traveled by the body is 2 times the Earth's radius. Let's assume the Earth's radius is R.

Therefore, the work done against the gravitational force is given by:

Work = force x distance
= mg x 2R
= 2mgR

Now, the time taken to reach the maximum height is given by:

t = (2u sinθ) / g

Where:
t = time taken
u = initial velocity
θ = angle of projection
g = acceleration due to gravity

Since the body is projected vertically upwards, the angle of projection is 90° and the sine of 90° is 1.

Therefore, the time taken to reach the maximum height is:

t = (2u) / g

Now, the power exerted by the gravitational force can be calculated using the formula:

Power = Work / time
= (2mgR) / [(2u) / g]
= mg^2R/u

From this equation, we can see that the power exerted by the gravitational force is inversely proportional to the initial velocity (u). As the body reaches the highest position, the velocity becomes zero. Therefore, the power exerted by the gravitational force is greatest at the highest position of the body.

Hence, the correct answer is option 'A' - at the highest position of the body.

$^{-1}$ and 2 kg second part moving with a velocity of 8 ms$^{-1}$. What is the mass and velocity of the third part?

To solve this problem, we need to use the law of conservation of momentum. According to this law, the total momentum of a system of objects remains constant if there are no external forces acting on the system. In other words, the sum of the momenta of all the objects before the explosion is equal to the sum of the momenta of all the objects after the explosion.

Before the explosion, the rock had zero velocity, so its momentum was zero. After the explosion, the two parts that went off at right angles to each other have the following momenta:

First part: momentum = mass x velocity = 1 kg x 12 ms$^{-1}$ = 12 kg ms$^{-1}$

Second part: momentum = mass x velocity = 2 kg x 8 ms$^{-1}$ = 16 kg ms$^{-1}$

The total momentum of these two parts is:

Total momentum = 12 kg ms$^{-1}$ + 16 kg ms$^{-1}$ = 28 kg ms$^{-1}$

According to the law of conservation of momentum, the momentum of the third part must be equal and opposite to the total momentum of the first two parts. Let's call the mass of the third part "m" and its velocity "v". Then we have:

Momentum of third part = -28 kg ms$^{-1}$

Momentum = mass x velocity

Therefore:

-mv = -28 kg ms$^{-1}$

Solving for "m", we get:

m = 28/v

Now we can use the law of conservation of energy to find the velocity of the third part. According to this law, the total kinetic energy of a system of objects remains constant if there are no external forces acting on the system. In other words, the sum of the kinetic energies of all the objects before the explosion is equal to the sum of the kinetic energies of all the objects after the explosion.

Before the explosion, the rock had zero kinetic energy, so its total kinetic energy was zero. After the explosion, the two parts that went off at right angles to each other have the following kinetic energies:

First part: KE = 0.5 x mass x velocity$^2$ = 0.5 x 1 kg x (12 ms$^{-1}$)$^2$ = 72 J

Second part: KE = 0.5 x mass x velocity$^2$ = 0.5 x 2 kg x (8 ms$^{-1}$)$^2$ = 64 J

The total kinetic energy of these two parts is:

Total KE = 72 J + 64 J = 136 J

According to the law of conservation of energy, the kinetic energy of the third part must be equal to the difference between the total kinetic energy of the first two parts and the initial kinetic energy of the rock. The initial kinetic energy of the rock was zero, so we have:

KE of third part = Total KE - 0 = 136 J

Using the formula for kinetic energy, we can write:

0.5mv$^2$ = 136 J

Solving for "v", we get:

v = $\sqrt{\frac{272}{m}}$

Substituting

Perpendicular to the original direction of B
d)Cannot be determined

Answer:
c) Perpendicular to the original direction of B

Explanation:
We can solve this problem using conservation of momentum and conservation of kinetic energy.

Before the collision, the total momentum of the system is:

p = m2v

Since sphere A is at rest, its momentum is zero.

The total kinetic energy of the system before the collision is:

K = (1/2)m2v^2

After the collision, the spheres move in different directions. Let the velocity of sphere A be u and the velocity of sphere B be w. Then, the total momentum of the system after the collision is:

p' = m1u + m2w

Since sphere B moves perpendicular to its original direction, we can write:

w = 2v

Using conservation of momentum, we have:

m2v = m1u + m2(2v)

Simplifying, we get:

u = (m2 - 2m1)v / m1

Now, using conservation of kinetic energy, we have:

(1/2)m1u^2 + (1/2)m2w^2 = (1/2)m2v^2

Substituting the values of u and w, we get:

(m2 - 2m1)v^2 = m1u^2

Simplifying, we get:

u = sqrt((m2 - 2m1)/m1) v

Since m1 and m2 are positive, (m2 - 2m1)/m1 is negative. Therefore, u is imaginary, which means that sphere A moves in a direction perpendicular to the original direction of sphere B. Hence, the answer is option c) Perpendicular to the original direction of B.

Given:
Velocity of water, v = 2 m/s
Mass per unit length of water, m = 100 kg/m

To find:
Power of the engine

Formula used:
Power = Force × Velocity

Force = mass × acceleration = mass × change in velocity / time

Change in velocity = final velocity - initial velocity = v - 0 = v

Time taken to move out of the pipe, t = length of the pipe / velocity of water

Let the length of the pipe be L.

Therefore, t = L / v

Force = m × v / t = m × v² / L

Power = Force × Velocity = m × v² / L × v = m × v² / L

Substituting the given values, we get:

Power = 100 kg/m × (2 m/s)² / L = 400 W

Therefore, the power of the engine is 400 W.

-1. The radius of the cylinder is 0.5 m. Calculate the kinetic energy of the cylinder.

First, we need to calculate the angular velocity of the cylinder, which can be found using the formula:

v = ωr

where v is the linear velocity, ω is the angular velocity, and r is the radius of the cylinder. Rearranging this formula, we get:

ω = v/r

Substituting the given values, we get:

ω = 4/0.5 = 8 rad/s

The kinetic energy of the cylinder can be calculated using the formula:

KE = (1/2)Iω^2 + (1/2)mv^2

where I is the moment of inertia of the cylinder, m is the mass of the cylinder, and v is the linear velocity.

The moment of inertia of a solid cylinder is given by:

I = (1/2)mr^2

Substituting the given values, we get:

I = (1/2)×3×(0.5)^2 = 0.375 kg m^2

Substituting all the values in the formula for KE, we get:

KE = (1/2)×0.375×8^2 + (1/2)×3×4^2
= 12 + 24
= 36 J

Therefore, the kinetic energy of the cylinder is 36 Joules.