Given in Figures 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 Figures 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.
$x>a ; 0$
Total energy of a system is given by the relation
$E= P.E. + K . E$
$- K.E. =E- P.E.$
Kinetic energy of a body is a positive quantity. It cannot be negative. Therefore, the particle will not exist in a region where K.E. becomes negative.
In the given case, the potential energy ( $V_{0}$ ) of the particle becomes greater than total energy ( $E$ ) for $x>a$. Hence, kinetic energy becomes negative in this region. Therefore, the particle will not exist is this region. The minimum total energy of the particle is zero.
All regions
In the given case, the potential energy $\left(V_{0}\right)$ is greater than total energy $(E)$ in all regions. Hence, the particle will not exist in this region.
$x>a$ and $x < b ;-V_{1}$
In the given case, the condition regarding the positivity of K.E. is satisfied only in the region between $x>a$ and $x < b$
The minimum potential energy in this case is $-V_{1} .$ Therfore, $K.E. =E-\left(-V_{1}\right)=E+V_{1}$ Therefore, for the positivity of the kinetic energy, the totaol energy of the particle must be greater than $-V_{1} .$ So, the minimum total energy the particle must have is $-V_{1}$
$-\frac{b}{2} < x < \frac{a}{2} ; \;\;\frac{a}{2} < x < \frac{b}{2} ;-V_{1}$
In the given case, the potential energy ( $V_{0}$ ) of the particle becomes greater than the total energy $(E)$ for $-\frac{b}{2} < x < \frac{b}{2}$ and $-\frac{a}{2} < x < \frac{a}{2}$. Therefore, the particle will not exist in these regions.
The minimum potential energy in this case is $-V_{1} .$ Therfore, $K.E. =E-\left(-V_{1}\right)=E+V_{1}$ Therefore, for the positivity of the kinetic energy, the totaol energy of the particle must be greater than $-V_{1} .$ So, the minimum total energy the particle must have is $-V_{1} .$
Similar Questions
Starting from rest on her swing at initial height $h_0$ above the ground, Saina swings forward. At the lowest point of her motion, she grabs her bag that lies on the ground. Saina continues swinging forward to reach maximum height $h_1$ . She then swings backward and when reaching the lowest point of motion again, she simple lets go off the bag, which falls freely. Saina's backward swing then reaches maximum height $h_2$ . Neglecting air resistance, how are the three heights related?
Starting from rest on her swing at initial height $h_0$ above the ground, Saina swings forward. At the lowest point of her motion, she grabs her bag that lies on the ground. Saina continues swinging forward to reach maximum height $h_1$ . She then swings backward and when reaching the lowest point of motion again, she simple lets go off the bag, which falls freely. Saina's backward swing then reaches maximum height $h_2$ . Neglecting air resistance, how are the three heights related?
A person trying to lose weight (dieter) lifts a $10\; kg$ mass, one thousand times, to a hetght 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?
$(b)$ 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?
A person trying to lose weight (dieter) lifts a $10\; kg$ mass, one thousand times, to a hetght 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?
$(b)$ 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?
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
A bullet weighing $10 \,g$ and moving with a velocity $300 \,m / s$ strikes a $5 \,kg$ block of ice and drop dead. The ice block is kept on smooth surface. The speed of the block after the collision is ........ $cm / s$
A bullet weighing $10 \,g$ and moving with a velocity $300 \,m / s$ strikes a $5 \,kg$ block of ice and drop dead. The ice block is kept on smooth surface. The speed of the block after the collision is ........ $cm / s$
A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be
A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be