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Q6.3:
Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.
Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.
Solution :
Theoretical Foundation: Energy Constraints in Potential Wells
The motion of a particle in one dimension is governed by the principle of Conservation of Mechanical Energy: $E = K + U(x)$, where $E$ is the constant total energy, $K$ is the kinetic energy, and $U(x)$ is the potential energy function. Since kinetic energy $K = \frac{1}{2}mv^2$ must always be non-negative ($K \geq 0$), it follows that $E - U(x) \geq 0$, or $E \geq U(x)$ [Per the Principle of Conservation of Energy].
Regions where $E < U(x)$ are "classically forbidden" regions because they would require $K < 0$, which is physically impossible for a particle of real mass. The minimum energy $E_{min}$ the particle must have corresponds to the local minimum value of $U(x)$.
Analysis of Potential Energy Curves
| Case | Forbidden Region ($E < U(x)$) | Minimum Energy ($E_{min}$) | Physical Context |
|---|---|---|---|
| (a) | $x > a$ | $E_{min} = 0$ | Repulsive wall (e.g., a hard elastic sphere hitting a wall). |
| (b) | $x < x_1$ and $x > x_2$ | $E_{min} = -V_0$ | Simple Harmonic Oscillator (e.g., mass on a spring). |
| (c) | $x < 0$ and $x > a$ | $E_{min} = 0$ | Particle in a box (e.g., gas molecule in a rigid container). |
| (d) | $x_1 < x < x_2$ (if $E < V_1$) | $E_{min} = -V_1$ | Double well potential (e.g., inversion of ammonia molecule). |
Visual Representation of a Potential Well
Granular Deductions
- Case (a): The potential rises monotonically. The particle is reflected at $x=a$. It cannot enter $x > a$ as $U(x) \to \infty$.
- Case (b): This is a symmetric parabolic well. The particle is trapped between two turning points $x_1$ and $x_2$ where $E = U(x)$. Outside these points, $U(x) > E$, rendering them forbidden.
- Case (c): A "square well" potential. The particle is strictly confined to $0 \leq x \leq a$. Outside this range, the potential is infinite, creating an impassable barrier.
- Case (d): The "double well" potential. If $E < V_1$, the particle is trapped in either the left well or the right well, as the central barrier $V_1$ acts as a forbidden transition region [Per Classical Mechanics].
Final Solution: The particle cannot exist in any region where the potential energy $U(x)$ exceeds its total energy $E$. The minimum energy $E_{min}$ is equal to the global minimum of the potential energy function in each case.
More Questions from Class 11 Physics Work EXERCISES
- Q5.1(a): The sign of work done by a force on a body is important to understand. State carefully if the following quantity is positive or negative: work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
- Q5.1(b): The sign of work done by a force on a body is important to understand. State carefully if the following quantity is positive or negative: work done by gravitational force in the above case,
- Q5.1(c): The sign of work done by a force on a body is important to understand. State carefully if the following quantity is positive or negative: work done by friction on a body sliding down an inclined plane,
- Q5.1(d): The sign of work done by a force on a body is important to understand. State carefully if the following quantity is positive or negative: work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
- Q5.1(e): The sign of work done by a force on a body is important to understand. State carefully if the following quantity is positive or negative: work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
- Q5.10: A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time (t) is proportional to
- Q5.11: A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by (F = -\hat{i} + 2\hat{j} + 3\hat{k}) N where (\hat{i}, \hat{j}, \hat{k}) are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?
- Q5.12: An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (electron mass = (9.11 \times 10^{-31}) kg, proton mass = (1.67 \times 10^{-27}) kg, 1 eV = (1.60 \times 10^{-19}) J).
- Q5.13: A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey ? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is (10 m s^{-1}) ?
- Q5.14: A molecule in a gas container hits a horizontal wall with speed (200 m s^{-1}) and angle (30^{\circ}) with the normal, and rebounds with the same speed. Is momentum conserved in the collision ? Is the collision elastic or inelastic ?
- Q5.15: A pump on the ground floor of a building can pump up water to fill a tank of volume (30 m^{3}) in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump ?
- Q5.16: Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following is a possible result after collision ?
- Q5.17: The bob A of a pendulum released from (30^{\circ}) to the vertical hits another bob B of the same mass at rest on a table as shown. How high does the bob A rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic.
- Q5.18: The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance ?
- Q5.19: A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of (0.05 kg s^{-1}). What is the speed of the trolley after the entire sand bag is empty ?
- Q5.2: A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (a) work done by the applied force in 10 s, (b) work done by friction in 10 s, (c) work done by the net force on the body in 10 s, (d) change in kinetic energy of the body in 10 s, and interpret your results.
- Q5.20: A body of mass 0.5 kg travels in a straight line with velocity (v = a x^{3/2}) where (a = 5 m^{-1/2} s^{-1}). What is the work done by the net force during its displacement from (x = 0) to (x = 2 m) ?
- Q5.21(a): The blades of a windmill sweep out a circle of area A. If the wind flows at a velocity (v) perpendicular to the circle, what is the mass of the air passing through it in time (t) ?
- Q5.21(b): The blades of a windmill sweep out a circle of area A. What is the kinetic energy of the air ?
- Q5.21(c): The blades of a windmill sweep out a circle of area A. Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that (A = 30 m^{2}), (v = 36 km/h) and the density of air is (1.2 kg m^{-3}). What is the electrical power produced ?
- Q5.22(a): A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. How much work does she do against the gravitational force ?
- Q5.22(b): A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. Fat supplies (3.8 \times 10^{7}J) of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?
- Q5.23(a): A family uses 8 kW of power. Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW?
- Q5.23(b): A family uses 8 kW of power. Compare this area to that of the roof of a typical house.
- Q5.3: Given are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.
- Q5.4: The potential energy function for a particle executing linear simple harmonic motion is given by (V(x) = \frac{kx^{2}}{2}), where (k) is the force constant of the oscillator. For (k = 0.5 N m^{-1}), the graph of (V(x)) versus (x) is shown. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches (x = \pm 2 m).
- Q5.5(a): Answer the following : The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
- Q5.5(b): Answer the following : Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why ?
- Q5.5(c): Answer the following : An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth ?
- Q5.5(d): In one case, a man walks 2 m carrying a mass of 15 kg on his hands. In another case, he walks the same distance pulling a rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater?
- Q5.6(a): Choose the correct alternative: When a conservative force does positive work on a body, the potential energy of the body increases/decreases/remains unaltered.
- Q5.6(b): Choose the correct alternative: Work done by a body against friction always results in a loss of its kinetic/potential energy.
- Q5.6(c): Choose the correct alternative: The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
- Q5.6(d): Choose the correct alternative: In an inelastic collision of two bodies, the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.
- Q5.7(a): State if the following statement is true or false. Give reasons for your answer. In an elastic collision of two bodies, the momentum and energy of each body is conserved.
- Q5.7(b): State if the following statement is true or false. Give reasons for your answer. Total energy of a system is always conserved, no matter what internal and external forces on the body are present.
- Q5.7(c): State if the following statement is true or false. Give reasons for your answer. Work done in the motion of a body over a closed loop is zero for every force in nature.
- Q5.7(d): State if the following statement is true or false. Give reasons for your answer. In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
- Q5.8(a): Answer carefully, with reasons : In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact) ?
- Q5.8(b): Answer carefully, with reasons : Is the total linear momentum conserved during the short time of an elastic collision of two balls ?
- Q5.8(c): Answer carefully, with reasons : For an inelastic collision, state whether the total kinetic energy is conserved during the short time of collision and whether the total linear momentum is conserved during the same period.
- Q5.8(d): Answer carefully, with reasons : If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic ? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).
- Q5.9: A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time (t) is proportional to
- Q6.1(a): The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
- Q6.1(b): The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (b) work done by gravitational force in the above case,
- Q6.1(c): The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (c) work done by friction on a body sliding down an inclined plane,
- Q6.1(d): The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
- Q6.1(e): The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
- Q6.10: A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time $t$ is proportional to
- Q6.11: A body constrained to move along the z-axis of a coordinate system is subject to a constant force $F$ given by $F = (-\hat{i} + 2\hat{j} + 3\hat{k}) N$ where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?
- Q6.12: An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (electron mass = $9.11 \times 10^{-31}$ kg, proton mass = $1.67 \times 10^{-27}$ kg, 1 eV = $1.60 \times 10^{-19}$ J).
- Q6.13: A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey ? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is $10 \text{ m s}^{-1}$ ?
- Q6.14: A molecule in a gas container hits a horizontal wall with speed $200 \text{ m s}^{-1}$ and angle $30^{\circ}$ with the normal, and rebounds with the same speed. Is momentum conserved in the collision ? Is the collision elastic or inelastic ?
- Q6.15: A pump on the ground floor of a building can pump up water to fill a tank of volume $30 \text{ m}^3$ in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump ?
- Q6.16: Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed $V$. If the collision is elastic, which of the following (Fig. 6.14) is a possible result after collision ?
- Q6.17: The bob $A$ of a pendulum released from $30^{\circ}$ to the vertical hits another bob $B$ of the same mass at rest on a table as shown in Fig. 6.15. How high does the bob $A$ rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic.
- Q6.18: The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance ?
- Q6.19: A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of $0.05 \text{ kg s}^{-1}$. What is the speed of the trolley after the entire sand bag is empty ?
- Q6.2(a): A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (a) work done by the applied force in 10 s,
- Q6.2(b): A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (b) work done by friction in 10 s,
- Q6.2(c): A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (c) work done by the net force on the body in 10 s,
- Q6.2(d): A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (d) change in kinetic energy of the body in 10 s, and interpret your results.
- Q6.20: A body of mass 0.5 kg travels in a straight line with velocity $v = ax^{3/2}$ where $a = 5 \text{ m}^{-1/2} \text{ s}^{-1}$. What is the work done by the net force during its displacement from $x = 0$ to $x = 2$ m ?
- Q6.21(a): The blades of a windmill sweep out a circle of area $A$. (a) If the wind flows at a velocity $v$ perpendicular to the circle, what is the mass of the air passing through it in time $t$ ?
- Q6.21(b): The blades of a windmill sweep out a circle of area $A$. (b) What is the kinetic energy of the air ?
- Q6.21(c): The blades of a windmill sweep out a circle of area $A$. (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that $A = 30 \text{ m}^2$, $v = 36 \text{ km/h}$ and the density of air is $1.2 \text{ kg m}^{-3}$. What is the electrical power produced ?
- Q6.22(a): A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much work does she do against the gravitational force ?
- Q6.22(b): A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (b) Fat supplies $3.8 \times 10^7 \text{J}$ of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?
- Q6.23(a): A family uses 8 kW of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW?
- Q6.23(b): A family uses 8 kW of power. (b) Compare this area to that of the roof of a typical house.
- Q6.4: The potential energy function for a particle executing linear simple harmonic motion is given by $V(x) = \frac{kx^2}{2}$, where $k$ is the force constant of the oscillator. For $k = 0.5 \text{ N m}^{-1}$, the graph of $V(x)$ versus $x$ is shown in Fig. 6.12. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches $x = \pm 2$ m.
- Q6.5(a): Answer the following : (a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
- Q6.5(b): Answer the following : (b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why ?
- Q6.5(c): Answer the following : (c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth ?
- Q6.5(d): Answer the following : (d) In Fig. 6.13(i) the man walks 2 m carrying a mass of 15 kg on his hands. In Fig. 6.13(ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater ?
- Q6.6(a): Underline the correct alternative : (a) When a conservative force does positive work on a body, the potential energy of the body...
- Q6.6(b): Underline the correct alternative : (b) Work done by a body against friction always results in a loss of its...
- Q6.6(c): Underline the correct alternative : (c) The rate of change of total momentum of a many-particle system is proportional to the...
- Q6.6(d): Underline the correct alternative : (d) In an inelastic collision of two bodies, the quantities which do not change after the collision are the...
- Q6.7(a): State if each of the following statements is true or false. Give reasons for your answer. (a) In an elastic collision of two bodies, the momentum and energy of each body is conserved.
- Q6.7(b): State if each of the following statements is true or false. Give reasons for your answer. (b) Total energy of a system is always conserved, no matter what internal and external forces on the body are present.
- Q6.7(c): State if each of the following statements is true or false. Give reasons for your answer. (c) Work done in the motion of a body over a closed loop is zero for every force in nature.
- Q6.7(d): State if each of the following statements is true or false. Give reasons for your answer. (d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
- Q6.8(a): Answer carefully, with reasons : (a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact) ?
- Q6.8(b): Answer carefully, with reasons : (b) Is the total linear momentum conserved during the short time of an elastic collision of two balls ?
- Q6.8(c): Answer carefully, with reasons : (c) What are the answers to (a) and (b) for an inelastic collision ?
- Q6.8(d): Answer carefully, with reasons : (d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic ? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).
- Q6.9: A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time $t$ is proportional to
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