Mechanics of a Point Mass

Appendices

Summary

• A body with mass m can be described by the model of a point mass as long as its spatial extensions are small compared to its distance to other bodies.

• The motion of a body is described by a trajectory $\boldsymbol{r}(t)$, which the body traverses in the course of time. Its momentary velocity is $\boldsymbol{v}(t)=\dot{\boldsymbol{r}}=\,\mathrm{d}\boldsymbol{r}/\,\mathrm{d}t$ and its acceleration is $a(t)=\,\mathrm{d}v/\,\mathrm{d}t=\,\mathrm{d}^{2}r/\,\mathrm{d}t^{2}$.

• Motions with $\boldsymbol{a}(t)=\boldsymbol{0}$ are called uniform straight-line motions. Magnitude and direction of the velocity are constant.

• For the uniform circular motion the magnitude $|\boldsymbol{a}(t)|$ is constant, but the direction of $\boldsymbol{a}(t)$ changes uniformly with the angular velocity $\boldsymbol{\omega}$.

• A force acting on a freely movable body causes an acceleration and therefore a change of its state of motion.

• A body is in an equilibrium state if the vector sum of all acting forces is zero. In this case it does not change its state of motion.

• The state of motion of a body with mass m and velocity v is defined by the momentum $\boldsymbol{p}=m\cdot\boldsymbol{v}$.

• The force F acting on a body is defined as $F=\,\mathrm{d}\boldsymbol{p}/\,\mathrm{d}t$ (2. Newton’s law).

• For two bodies with masses m ₁ and m ₂ which interact with each other but not with other bodies the 3. Newtonian law is valid: $F_{1}=-F_{2}$ (F ₁ is the force acting on m ₁, F ₂ acting on $m_{2})$.

• The work executed by the force $\boldsymbol{F}(\boldsymbol{r})$ on a body moving along the trajectory $\boldsymbol{r}(t)$ is the scalar quantity $W=\int\boldsymbol{F}(\boldsymbol{r})\,\mathrm{d}\boldsymbol{r}$.

• Force fields where the work depends only on the initial point P ₁ and the final point P ₂ but not on the choice of the path between P ₁ and P ₂ are called conservative. For such fields is $\mathbf{rot}\,\boldsymbol{F}=\boldsymbol{0}$. All central force fields are conservative.

• To each point P in a conservative force field a potential energy $E_{\mathrm{p}}(P)$ can be attributed. The work $\int F(r)\,\mathrm{d}r=E(P_{1})-E(P_{2})$ executed on a body to move it from P ₁ to P ₂ is equal to the difference of the potential energies in P ₁ and P ₂. The choice of the point of zero energy is arbitrarily. Often one chooses $E(r=0)=0$ or $E(r=\infty)=0$.

• The potential energy $E(P)$ and the force $F(r)$ in a conservative force field are related by $\boldsymbol{F}(\boldsymbol{r})=-\mathbf{grad}E_{\mathrm{p}}$.

• The kinetic energy of a mass m moving with the velocity v is $E_{\mathrm{kin}}=\tfrac{1}{2}mv^{2}$.

• In a conservative force field the total energy $E=E_{p}+E_{\mathrm{kin}}$ is constant (law of energy conservation).

• The angular momentum of a mass m with momentum p, referred to the origin of the coordinate system is $\boldsymbol{L}=\boldsymbol{r}\times\boldsymbol{p}=m\cdot(\boldsymbol{r}\times\boldsymbol{v})$. The torque acting on a body in a force field $\boldsymbol{F}(\boldsymbol{r})$ is $\boldsymbol{D}=\boldsymbol{r}\times\boldsymbol{F}$. It is $\boldsymbol{D}=\,\mathrm{d}\boldsymbol{L}/\,\mathrm{d}t$.

• All planets of our solar system move in the central force field $F(r)=-G\cdot(m\cdot M/r^{2})\hat{\boldsymbol{r}}$ of the sun. Therefore their angular momentum is constant. Their motion is planar. Their trajectories are ellipses with the sun in one focal point.

• The gravitational field of extended bodies depends on the mass distribution. For spherical symmetric mass distributions with radius R the force field outside the body ($r> R$) is exactly that of a point mass, inside the body ($r<R$) the force $F(r)$ increases for homogeneous distributions linearly with r from $\boldsymbol{F}=0$ at the centre r = 0 to the maximum value at $r=R$.

• The free fall acceleration g of a body with mass m equals the gravitational field strength $\boldsymbol{G}=\boldsymbol{F}/m$ at the surface $r=R$ of the earth with mass M. With Newton’s law of gravity g can be expressed as $\boldsymbol{g}=G\cdot(M/R^{2})\hat{\boldsymbol{r}}$ ($G=$ gravitational constant). It can be determined from the measured oscillation period $T=2\pi\sqrt{L/g}$ of a pendulum with length L, or with gravitational balances.

Problems

2.1

A car drives on a road behind a foregoing truck (length of 25 $\mathrm{m}$) with a constant safety distance of 40 $\mathrm{m}$ and a constant velocity of 80 $\mathrm{k}$ $\mathrm{m}$/$\mathrm{h}$. As soon as the driver can foresee a free distance of 300 $\mathrm{m}$ he starts to overtake. Therefore he accelerates with $a=1.3\,\mathrm{m}/\mathrm{s}^{2}$ until he reaches a velocity of $v=100\,\mathrm{k}\mathrm{m}/\mathrm{h}$. Can he safely overtake? How long are time and path length of the overtaking procedure if he considers the same safety distance after the overtaking? Draw for illustration a diagram for $s(t)$ and $v(t)$.

2.2

A car drives half of a distance x with the velocity $v_{1}=80\,\mathrm{k}\mathrm{m}/\mathrm{h}$ and the second half with $v_{2}=40\,\mathrm{k}\mathrm{m}/\mathrm{h}$. Estimate and calculate the mean velocity $\overline{v}$ as the function of v ₁ and v ₂. Make the same consideration if $x_{1}=1/3x$ and $x_{2}=2/3x$.

2.3

A body moves with constant acceleration along the x-axis. It passes the origin x = 0 with $v=6\,\mathrm{c}\mathrm{m}/\mathrm{s}$. 2 $\mathrm{s}$ later it arrives at $x=10\,\mathrm{c}\mathrm{m}$. Calculate magnitude and direction of the acceleration.

2.4

An electron is emitted from the cathode with a velocity v ₀ and experiences in an electric field over a distance of 4 $\mathrm{c}$ $\mathrm{m}$ a constant acceleration $a=3\cdot 10^{14}\,\mathrm{m}/\mathrm{s}^{2}$, reaching a velocity of $7\cdot 10^{6}\,\mathrm{m}/\mathrm{s}$. How large was v ₀?

2.5

A body is thrown from a height $h=15\,\mathrm{m}$ with an initial velocity $v_{0}=5\,\mathrm{m}/\mathrm{s}$

• a) upwards,

• b) downwards.

Calculate for both cases the time until it reaches the ground.

• i Derive Eq. 2.13.

2.6

Give examples where both the magnitude and the direction of the acceleration are constant but the body moves nevertheless not on a straight line. Which conditions must be fulfilled for a straight line?

2.7

A car crashes with a velocity of 100 $\mathrm{k}$ $\mathrm{m}$/$\mathrm{h}$ against a thick tree. From which heights must it fall down in order to experience the same velocity when reaching the ground? Compare this with two equal cars with velocities of 100 $\mathrm{k}$ $\mathrm{m}$/$\mathrm{h}$ crashing head on against each other.

2.8

• a) A body moves with constant angular velocity $\omega=3\,\mathrm{rad}/\mathrm{s}$ on a vertical circle in the x-z-plane with radius $R=1\,\mathrm{m}$ in the gravity field $F=\{0,0,-g\}$ of the earth. How large are its velocities at the lowest and the highest point on the circle? How large is the difference between the two values? Could you relate this to the potential energy?

• b) A body starts with $v_{0}=0$ from the point $A(z=h)$ in Fig. 2.64 on the frictionless looping path. How large are velocities and accelerations in the points B and C of the circular path with radius R? What is the maximum ratio $R/h$ to prevent that the body falls down in B? How large is then the velocity $v(B)$?

  Fig. 2.64

  

  Looping path (Probl. 2.8 b)

2.9

How large is the escape velocity

• a) of the moon ($d=384\,000\,\mathrm{k}\mathrm{m}$) in the gravitational field of the earth?

• b) of a body on the surface of the moon in the gravitational field of the moon?

2.10

What is the minimum fuel mass of a one stage rocket with a payload of 500 $\mathrm{k}$ $\mathrm{g}$ for a horizontal launch at the equator to bring the rocket to the first escape velocity of $v_{1}=7.9\,\mathrm{k}\mathrm{m}/\mathrm{s}$ when the velocity of the propellant gas relative to the rocket is $v_{\mathrm{e}}=4.5\,\mathrm{k}\mathrm{m}/\mathrm{h}$

• a) in the east direction

• b) in the west direction?

2.11

Check the energy conservation law for the examples given in the text. Show, that () follows directly from the condition $E_{\mathrm{kin}}\geq E_{\mathrm{p}}$, i. e. $\tfrac{1}{2}mv^{2}\geq m\cdot g\cdot R$.

2.12

A rocket to the moon is launched from a point at the equator. How much energy is saved compared to a vertical launch, when it is shot in the eastern direction under $30^{\circ}$ against the horizontal?

2.13

A wooden cylinder (radius $r=0.1\,\mathrm{m}$, heights $h=0.6\,\mathrm{m}$) is vertically immersed in water with 2/3 of its length which is its equilibrium position. Which work has to be performed when it is pulled out of the water? How is the situation if the cylinder lies horizontally in the water? How deep does it immerse?

2.14

A body with mass $m=0.8\,\mathrm{k}\mathrm{g}$ is vertically thrown upwards. In the heights $h=10\,\mathrm{m}$ its kinetic energy is $200\,\mathrm{J}$. What is the maximum heights it can reach?

2.15

A spiral spring of steel with length $L_{0}=0.8\,\mathrm{m}$ is expanded by the force $F=20\,\mathrm{N}$ to a length $L=0.85\,\mathrm{m}$. Which work is needed to expand the spring to twice its initial length, if the force is always proportional to the expansion $\Delta L=L-L_{0}$?

2.16

What is the minimum initial velocity of a body at a vertical launch from the earth when it should reach the moon?

2.17

What is the distance of a geo-stationary satellite from the centre of the earth? Which energy is needed to launch it? How accurate has its distance to the earth centre be stabilized in order to maintain its position relative to a point on earth within $0.1\,\mathrm{k}\mathrm{m}/\mathrm{d}$?

2.18

What is the change of potential, kinetic and total energy of a satellite when its radius r on a stable circular orbit around the earth centre is changed? What is the ratio $E_{\mathrm{kin}}/E_{\mathrm{p}}$? Does it depend on r? Express the total energy E by m, g, r and the mass $M_{\mathrm{E}}$ of the earth. Are these quantities sufficient or are more needed?

2.19

Prove, that the force $\boldsymbol{F}=m\cdot g\cdot\sin\varphi\cdot\boldsymbol{e}_{\mathrm{t}}$ for the mathematical pendulum is conservative and that for arbitrary values of $\varphi$ conservation of energy $E_{\mathrm{kin}}+E_{\mathrm{p}}=\mathrm{const}$ holds.

2.20

Assume one is able to measure the length $L=10\,\mathrm{m}$ of a pendulum within $0.1\,\mathrm{m}\mathrm{m}$ and the period T within $10\,\mathrm{m}\mathrm{s}$. How many oscillation periods have to be measured in order to equalize the contribution of $\Delta L$ and $\Delta T$ to the accuracy of g? How large is then the uncertainty of g?

2.21

How much accuracy is gained for the determination of G with the gravity balance if the large masses M are increased by a factor of 10? How accurate has the measurement of the angle $\varphi$ to be in order to determine G with an accuracy of $10^{-4}$? Give some physical reasons for the limits of the accuracy of $\varphi$.

2.22

The comet Halley has a period of 76 years. His smallest distance to the sun is 0.59 AU. How large is its maximum distance to the sun and what is the eccentricity of its elliptical orbit? Hint: Look for a relation between T and $r_{\mathrm{min}}=a(1-\varepsilon)$ and $r_{\mathrm{max}}=a(1+\varepsilon)$.

2.23

Assume that the gravity acceleration at the equator of a rotating planet is $11.6\,\mathrm{m}/\mathrm{s}^{2}$, the centripetal acceleration $a=0.3\,\mathrm{m}/\mathrm{s}^{2}$ and the escape velocity for a vertical launch $23.6\,\mathrm{m}/\mathrm{s}$. At the heights $h=5000\,\mathrm{k}\mathrm{m}$ above the surface is $g=8.0\,\mathrm{m}/\mathrm{s}^{2}$. What are the radius R and the mass M of the planet. How fast is it rotating? Which planet meets these requirements?

2.24

The gravitational force exerted by the sun onto the moon is about twice as large as that exerted by the earth. Why is the moon still circling around the earth and has not escaped?

2.25

Which oscillation period would a pendulum have on the moon, if its period on the earth is 1 $\mathrm{s}$?

2.26

A vertical straight tunnel is cut through the earth between opposite points A to B on the earth surface.

• a) Show that without friction a body released in A performs a harmonic oscillation between A and B.

• b) What is the oscillation period?

• c) Compare this value with the period of a satellite, which circles around the earth closely above the surface.

• d) A straight tunnel is cut between London and New York. What is the travel time of a train without friction and extra driving force (besides gravity) which starts in London with the velocity $v_{0}=0$? How much does the time change, if $v_{0}=40\,\mathrm{m}/\mathrm{s}$?

2.27

Calculate the distance earth-moon from the period of revolution of the moon $T=27\,\mathrm{d}$ (mass of the earth is $M=6\cdot 10^{24}\,\mathrm{k}\mathrm{g}$).

2.28

Saturn has a mass $M=5.7\cdot 10^{26}\,\mathrm{k}\mathrm{g}$ and a mean density of $0.71\,\mathrm{g}/\mathrm{c}\mathrm{m}^{3}$. How large is the gravitational acceleration on its surface?

2.29

How large is the relative change of the gravity acceleration g between a point on the earth surface and a point with $h=160\,\mathrm{k}\mathrm{m}$ above the surface?

2.30

How large is the change $\Delta g$ of the earth acceleration due to the attraction by

• a) the moon and

• b) the sun?

Compare the two changes and discuss them. How large is the relative change $\Delta g/g$?

2.31

Two spheres made of lead with masses $m_{1}=m_{2}=20\,\mathrm{k}\mathrm{g}$ are suspended by two thin wires with length $L=100\,\mathrm{m}$ where the suspension points are $0.2\,\mathrm{m}$ apart. What is the distance between the centres of the spheres, when the gravitational field of the earth is assumed to be spherical symmetric?

• a) without

• b) with the gravitational force between the two masses.

2.32

Based on the energy conservation law determine the velocity of the earth in its closest distance from the sun (Perihelion) and for the largest distance (aphelion). How large is the difference $\Delta v$ to the mean velocity? Discuss the relation between the eccentricity of the elliptical orbit and $\Delta v$.

2.33

A satellite orbiting around the earth has the velocity $v_{A}=5\,\mathrm{k}\mathrm{m}/\mathrm{s}$ in the aphelion and $v_{P}=7\,\mathrm{k}\mathrm{m}/\mathrm{s}$ in its perihelion. How large are minor and major half axes of its elliptical orbit?

2.34

Prove the equation (2.78a).