NEET UG Question of the Day

Assertion (A) is true. According to Newton's second law for a system of particles, the net external force (F_ext) acting on the system is equal to the product of the total mass (M) of the system and the acceleration of its center of mass (a_CM), i.e., F_ext = M * a_CM. If the net external force (F_ext) is zero, then the acceleration of the center of mass (a_CM) must be zero. Since acceleration is the rate of change of velocity, a zero acceleration implies that the velocity of the center of mass (V_CM) is constant.

Reason (R) is also true. The principle of conservation of linear momentum states that if the net external force acting on a system is zero (i.e., an isolated system), the total linear momentum of the system remains constant. This is a fundamental principle derived from Newton's laws. Furthermore, Reason (R) is the correct explanation for Assertion (A). The total linear momentum (P_total) of a system of particles is defined as the product of the total mass (M) of the system and the velocity of its center of mass (V_CM), i.e., P_total = M * V_CM. If, as stated in Reason (R), the total linear momentum (P_total) is conserved (constant) for an isolated system, and the total mass (M) of the system is constant (assuming no mass exchange with the surroundings), then the velocity of the center of mass (V_CM) must necessarily be constant. Thus, the conservation of total linear momentum directly explains why the velocity of the center of mass remains constant when the net external force is zero.

The generalized work-energy theorem states that the work done by all non-conservative forces (W_nc) acting on a system is equal to the change in its mechanical energy (ΔE_mechanical). This can be expressed as W_nc = ΔE_mechanical = ΔKE + ΔPE.

In this problem, the system is the block. The conservative forces acting on the block are gravity and the normal force. Since the motion is strictly horizontal, the work done by gravity and the normal force is zero. The non-conservative forces are the applied force 'F' and the kinetic friction force 'f_k'.

1. **Work done by the applied force (F):** Since the force is constant and in the direction of displacement, the work done by F is W_F = Fd.

2. **Work done by kinetic friction (f_k):** The kinetic friction force opposes the motion. Its magnitude is f_k = μN. On a horizontal surface, the normal force N equals the gravitational force mg. So, f_k = μmg. The work done by friction is W_friction = -f_k * d = -μmgd. The negative sign indicates that the friction force opposes the displacement.

3. **Net work done by non-conservative forces (W_nc):** This is the sum of the work done by the applied force and the friction force. W_nc = W_F + W_friction = Fd - μmgd = (F - μmg)d.

4. **Change in mechanical energy (ΔE_mechanical):** According to the generalized work-energy theorem, ΔE_mechanical = W_nc. Therefore, ΔE_mechanical = (F - μmg)d. Since the motion is horizontal, the change in potential energy (ΔPE) is zero, so ΔE_mechanical = ΔKE + 0 = ΔKE. This means the change in kinetic energy is also (F - μmg)d.

Based on this analysis:

• The statement 'Work done by net non-conservative forces = (F - μmg)d; Change in mechanical energy = (F - μmg)d' accurately reflects both the calculated net non-conservative work and its direct relationship to the change in mechanical energy.
• The statement 'Work done by net non-conservative forces = -μmgd; Change in mechanical energy = (F - μmg)d' is incorrect because the applied force 'F' is also a non-conservative force in this context (as it's not derived from a potential energy of the system itself) and its work (Fd) must be included in the net non-conservative work.
• The statement 'Work done by net non-conservative forces = (F - μmg)d; Change in mechanical energy = 0' is incorrect. Mechanical energy is not conserved when non-conservative forces do a net non-zero work (assuming F ≠ μmg), which is the case here. The change in mechanical energy must be equal to the net non-conservative work.
• The statement 'Work done by net non-conservative forces = (F + μmg)d; Change in mechanical energy = (F + μmg)d' is incorrect because the work done by kinetic friction is always negative as it opposes motion, not positive.

Assertion (A) is true. According to Newton's second law for a system of particles, the net external force (F_ext) acting on the system is equal to the product of the total mass (M) of the system and the acceleration of its center of mass (a_CM), i.e., F_ext = M * a_CM. If the net external force (F_ext) is zero, then the acceleration of the center of mass (a_CM) must be zero. Since acceleration is the rate of change of velocity, a zero acceleration implies that the velocity of the center of mass (V_CM) is constant.

Reason (R) is also true. The principle of conservation of linear momentum states that if the net external force acting on a system is zero (i.e., an isolated system), the total linear momentum of the system remains constant. This is a fundamental principle derived from Newton's laws. Furthermore, Reason (R) is the correct explanation for Assertion (A). The total linear momentum (P_total) of a system of particles is defined as the product of the total mass (M) of the system and the velocity of its center of mass (V_CM), i.e., P_total = M * V_CM. If, as stated in Reason (R), the total linear momentum (P_total) is conserved (constant) for an isolated system, and the total mass (M) of the system is constant (assuming no mass exchange with the surroundings), then the velocity of the center of mass (V_CM) must necessarily be constant. Thus, the conservation of total linear momentum directly explains why the velocity of the center of mass remains constant when the net external force is zero.

The generalized work-energy theorem states that the work done by all non-conservative forces (W_nc) acting on a system is equal to the change in its mechanical energy (ΔE_mechanical). This can be expressed as W_nc = ΔE_mechanical = ΔKE + ΔPE.

In this problem, the system is the block. The conservative forces acting on the block are gravity and the normal force. Since the motion is strictly horizontal, the work done by gravity and the normal force is zero. The non-conservative forces are the applied force 'F' and the kinetic friction force 'f_k'.

1. **Work done by the applied force (F):** Since the force is constant and in the direction of displacement, the work done by F is W_F = Fd.

2. **Work done by kinetic friction (f_k):** The kinetic friction force opposes the motion. Its magnitude is f_k = μN. On a horizontal surface, the normal force N equals the gravitational force mg. So, f_k = μmg. The work done by friction is W_friction = -f_k * d = -μmgd. The negative sign indicates that the friction force opposes the displacement.

3. **Net work done by non-conservative forces (W_nc):** This is the sum of the work done by the applied force and the friction force. W_nc = W_F + W_friction = Fd - μmgd = (F - μmg)d.

4. **Change in mechanical energy (ΔE_mechanical):** According to the generalized work-energy theorem, ΔE_mechanical = W_nc. Therefore, ΔE_mechanical = (F - μmg)d. Since the motion is horizontal, the change in potential energy (ΔPE) is zero, so ΔE_mechanical = ΔKE + 0 = ΔKE. This means the change in kinetic energy is also (F - μmg)d.

Based on this analysis:

• The statement 'Work done by net non-conservative forces = (F - μmg)d; Change in mechanical energy = (F - μmg)d' accurately reflects both the calculated net non-conservative work and its direct relationship to the change in mechanical energy.
• The statement 'Work done by net non-conservative forces = -μmgd; Change in mechanical energy = (F - μmg)d' is incorrect because the applied force 'F' is also a non-conservative force in this context (as it's not derived from a potential energy of the system itself) and its work (Fd) must be included in the net non-conservative work.
• The statement 'Work done by net non-conservative forces = (F - μmg)d; Change in mechanical energy = 0' is incorrect. Mechanical energy is not conserved when non-conservative forces do a net non-zero work (assuming F ≠ μmg), which is the case here. The change in mechanical energy must be equal to the net non-conservative work.
• The statement 'Work done by net non-conservative forces = (F + μmg)d; Change in mechanical energy = (F + μmg)d' is incorrect because the work done by kinetic friction is always negative as it opposes motion, not positive.

The horizontal range (R) of a projectile launched with an initial speed 'u' at an angle 'θ' with the horizontal is given by the formula:

R = (u^2 * sin(2θ)) / g

Given values are: Initial speed (u) = 10 m/s, launch angle (θ) = 30 degrees, and acceleration due to gravity (g) = 10 m/s^2.

Substitute these values into the range formula:

R = (10^2 * sin(2 * 30°)) / 10

R = (100 * sin(60°)) / 10

We know that sin(60°) = sqrt(3)/2 ≈ 0.866.

R = (100 * 0.866) / 10

R = 10 * 0.866

R = 8.66 meters

Rounding this to one decimal place, the approximate maximum horizontal distance covered is 8.7 meters.

Distractor Analysis:

• A value of 5.0 meters would be obtained if the student incorrectly used sin(θ) instead of sin(2θ) in the range formula. This would lead to R = (100 * sin(30°)) / 10 = (100 * 0.5) / 10 = 5 meters.
• A value of 1.25 meters represents the maximum height (H) reached by the droplet, calculated using H = (u^2 * sin^2(θ)) / (2g) = (100 * (sin(30°))^2) / (2 * 10) = (100 * (0.5)^2) / 20 = (100 * 0.25) / 20 = 25 / 20 = 1.25 meters. This is a common confusion between range and maximum height.
• A value of 10.0 meters would result if the student incorrectly assumed sin(2θ) = 1 (which only occurs for a launch angle of 45 degrees), leading to R = u^2/g = 10^2/10 = 10 meters.