Questions and Answers of Mechanics

An industrial electromechanical device (Fig. 6.9) is used to stop small objects in motion that have become negatively charged by electrostatic friction during production. The motion of such objects
A material point of mass \(m=500 \mathrm{~g}\) is suspended from a fixed point \(\mathrm{O}\) by an inextensible wire of length \(L=50 \mathrm{~cm}\). The material point, initially in an equilibrium
In the picture of Fig. 6.10, you see in action a fundamental force of nature (called the Lorentz force) acting on particles with electric charge \(q\). The Lorentz force is expressed by the relation
Estimate the mass \(M_{T}\) of the Earth using the density function given in Fig. 7.1. Assume constant density values in the regions of inner core, outer core, mantle, crust. Compare with the
Estimate the number of atoms in a fine grain of sand, knowing that its diameter is about \(20 \mu \mathrm{m}\) and that the atomic distances are of the order of \(1 \AA=10^{-10} \mathrm{~m}\).
Determine the C.M. position of the rod of Fig. 7.6 in the case where the linear density \(\lambda\) of the rod varies with length according to the relation \(\lambda(x)=\lambda_{0}
Determine the C.M. position of the rod of Fig. 7.6 in the case when its cross section in the \(y z\) plane is circular as in Fig. 7.7 (right) and with surface density \(\sigma\) of the rod cross
An isolated system is made by two bodies of mass \(m_{1}=12.0 \mathrm{~kg}\) and \(m_{2}=2.0 \mathrm{~kg}\). They are initially in equilibrium and are separated by a distance \(d=1.23 \mathrm{~cm}\),
A person of mass \(m=67 \mathrm{~kg}\) stands on a boat of mass \(M=420 \mathrm{~kg}\). The boat can move, without friction, on the surface of a calm lake. The system is initially in equilibrium. The
A railway carriage of mass \(m=35.6\) ton moves with speed \(v_{0}=2.10 \mathrm{~m} / \mathrm{s}\) and hits, remaining attached to them, three other identical carriages, joined together and moving in
A bullet of mass \(m=120 \mathrm{~g}\) strikes horizontally, with velocity \(v_{0}\), a body of mass \(M=250 \mathrm{~g}\), hanging vertically from a pin by an ideal cable of length \(L=1.2
A neutron has a kinetic energy such that its initial velocity is \(\mathbf{v}_{i}^{n}=610^{5} \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}\). It strikes a proton, a particle which can be assumed to have
A firework of mass \(M=0.3 \mathrm{~kg}\) is designed to break into two fragments. It is launched with a velocity of magnitude \(v_{0}=60 \mathrm{~m} / \mathrm{s}\) in a direction that forms an angle
A homogeneous hemisphere, mass \(m=170 \mathrm{~g}\) and radius \(R=3.0 \mathrm{~cm}\), is placed on a horizontal plane and it is pulled by a horizontal force \(\mathbf{F}\), of unknown magnitude,
Two carts connected to each other (Fig. 7.16) with a horizontal inextensible bar, move with a constant velocity \(v_{0}=10.0 \mathrm{~m} / \mathrm{s}\) on straight track parallel to the ground. Each
A ball of mass \(m\) and negligible size can slide without friction on the inner wall of a hemisphere of radius \(R=15 \mathrm{~cm}\), as shown in Fig. 7.17. The \(z\) axis shown in figure represents
Two identical masses \(m=0.25 \mathrm{~kg}\) are suspended from a vertical rod by two rigid bars of length \(L=20 \mathrm{~cm}\) and negligible mass (Fig. 7.17, ). When the system rotates around the
A body with a mass of \(m=2.5 \mathrm{~kg}\) (including a small amount of negligible mass of explosive) is thrown vertically upwards with an initial velocity of magnitude \(v_{0}\). When it reaches
A simple estimate of the intensity of an impulsive force can be obtained by dropping a tennis ball (mass \(m=58 \mathrm{~g}\), radius \(R=32.5 \mathrm{~mm}\) ) with a rigid floor. If dropped at rest
A hollow cylinder is closed at the top end by a movable piston of mass \(M\) that can slide in the hollow part of the cylinder without friction. A flow of gas equal to \(\Phi=\frac{\Delta M_{\text
A mechanical system consists of a cubic block of mass \(M\) and a spring of elastic constant \(k\) and negligible mass rigidly anchored above the block. The cube \(M\) is stationary on a horizontal
A rocket, in the absence of gravity and other external forces, expels gas at a speed \(v_{g}=1.5\) \(\mathrm{km} / \mathrm{s}\) relative to the engines, starting from a standstill. What is its speed
If \(v_{g}\) is the speed of the ejected gases relative to the engine of a rocket, determine the ratio of the initial mass to the final mass of the rocket to obtain a final rocket speed of \(v_{f}=10
Compare the performance of a one-stage rocket with that of a two-stage rocket having similar fuel mass and structure mass. Rocket \(A\) has mass \(M_{A}=11 \mathrm{t}\), of which \(m_{A}^{c}=9.7
In Table 8.1 are reported half lives of nuclei with values greater than \(10^{15}\) years, while the current age of the Universe is only \(1.37 \times 10^{10}\) years. Could you explain how it is
A projectile particle has mass \(m_{a}\) and that target mass \(m_{b}\). Determine under what conditions, in a perfectly elastic collision, there is maximum energy transfer between the projectile and
Show that the relationship between half-life \(t_{1 / 2}\) and mean life \(\tau\) is given by the relation 11/2 = 7 In(2)
A small ball of mass \(M=800 \mathrm{~g}\) and negligible size lies on a smooth horizontal plane and is initially attached to the end of a spring of spring constant \(k\) so as to compress it by a
Consider a sled of mass \(M=10 \mathrm{~kg}\) and length \(L=1.0 \mathrm{~m}\) on which is placed a mass \(m=3 \mathrm{~kg}\) (considered point-like) that can slide without friction in the plane of
A body of mass \(m=10.0 \mathrm{~g}\) is dropped from an initial height \(h=50.0 \mathrm{~cm}\) along a frictionless circular guide, as in Fig. 8.15. At the end of the circular guide, the body moves
A device schematized in Fig.8.16 consists of: two equal balls of mass \(M=40 \mathrm{~g}\), constrained to move without any friction along a straight guide, connected by a spring of elastic constant
There are the three particles shown in the Fig. 8.16, with \(m=1.00 \mathrm{~kg}\) and \(M=2 m\). The three particles are aligned in the direction of their centers, and no sources of friction are
A ball of mass \(m=0.200 \mathrm{~kg}\) is launched from ground level, with velocity \(v_{0}\) and an angle of elevation \(\alpha=65^{\circ}\), toward a smooth wall, perpendicular to the trajectory,
The mole is the unit for measuring the amount of substance, and is defined as the amount of substance that contains \(6.0225 \times 10^{23}\) elementary entities. This corresponds to the numerical
With the data from the previous question, how many water molecules are there in one \(\mathrm{cm}^{3}\) (under standard pressure and temperature conditions)?Previous QuestionThe mole is the unit for
Determine the number \(n\) of atoms per \(\mathrm{cm}^{3}\) present in gold \((Z=79, A=197\), density 19.3 \(\mathrm{g} / \mathrm{cm}^{3}\) ) and estimate the distance between the centers of two
In an experiment in which helium nuclei are sent against a target consisting of \(1 \mathrm{~mm}\) of gold, it is observed that \(99.9 \%\) reaches a detector that is on the beam line. Assuming that
The mass of an atom is concentrated in the nucleus. The nuclear radius is expressible in terms of the mass number \(A\) by the relation \(R=1.2 A^{1 / 3} \mathrm{fm}\), where \(1 \mathrm{fm}=10^{-15}
The density of the interstellar medium in our Galaxy is measured to be \(10^{-21} \mathrm{~kg} \mathrm{~m}^{-3}\) and consists mainly of hydrogen. Estimate how many hydrogen atoms per
The Galaxy can be schematized as a disk of radius \(15 \mathrm{kpc}\) and thickness \(200 \mathrm{pc}\). Since we know that \(1 \mathrm{pc}=3.08 \times 10^{16} \mathrm{~m}\), determine the volume of
A glass filled with water has radius \(3 \mathrm{~cm}\); left open, in \(4 \mathrm{~h}\) the level has dropped \(1 \mathrm{~mm}\). Determine, in grams/hour, the rate at which water evaporates. Also
A beam sending neutrinos from CERN, Geneva-Switzerland, to the Gran Sasso Laboratories in Italy, near L'Aquila, was operational between 2006 and 2012. The distance as the crow flies (i.e., along
Looking at the relationship (1.12), determine the percent error made using the approximation \(\sin \theta=\theta\) for angles equal to \(10^{\circ}, 5^{\circ}\) and \(1^{\circ}\). sin xx - - 6 +0(x)
Three fundamental magnitudes connecting quantities defined in mechanics are: \((i)\) the speed of light in vacuum, \(c=299792458 \mathrm{~m} \mathrm{~s}^{-1}\); (ii) Planck's reduced constant,
Still making use of the three quantities introduced in the question 13, using dimensional analysis get the quantity with the dimensions of a time, \(t_{P}\), called Planck time. Express \(t_{P}\) in
Still making use of the three quantities introduced in the question 13, using dimensional analysis get the quantity with the dimensions of a length, \(l_{P}\), called Planck's length. Express
The law of radioactive decay is given by the relation \(N(t)=N_{o} e^{-\lambda t}\), where \(N_{o}\) is the initial number of nuclei of a certain substance, \(N(t)\) those remaining after a certain
What relation must be valid between the vectors \(\mathbf{a}\) and \(\mathbf{b}\), which are different from each other and nonzero, so that the relation: \((\mathbf{a}+\mathbf{b})
Show that if the magnitudes of the sum and difference between two vectors are equal, then the vectors are perpendicular to each other.
In a Cartesian reference system two vectors are defined as \(\mathbf{a}=2 c \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\mathbf{b}=4 \hat{\mathbf{i}}+\) \((c-1) \hat{\mathbf{j}}+5
Show that if the sum and difference between two vectors are perpendicular, then the magnitudes of the two vectors are equal.
Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are equal in magnitude. Their sum has magnitude 4 and their vector product magnitude 16.Determine the magnitude of the two vectors.
Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) comply with the following conditions: \((i) \mathbf{a} \cdot \mathbf{b}=20 ;(\) ii \()(\mathbf{a}+\mathbf{b}) \cdot \mathbf{a}=36\); (iii)
A displacement \(\mathbf{s}\) of magnitude \(3 \mathrm{~m}\) is made in a Cartesian system at an angle \(\theta=30^{\circ}\) with the \(x\)-axis. Express the vector in Cartesian coordinates.
Find the vector \(\mathbf{c}\) which added to the vectors \(\mathbf{a}=(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) and \(\mathbf{b}=(2
Given the two vectors \(\mathbf{a}\) and \(\mathbf{b}\) of the Question 8 express their sum, difference, scalar product and vector product in Cartesian coordinates. What is the angle \(\alpha\)
Given two vectors \(\mathbf{a}\) and \(\mathbf{b}\), show that in intrinsic representation their vector product \(\mathbf{A}=\mathbf{a} \times \mathbf{b}\) corresponds to the oriented area of the
Given three vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) show that in intrinsic representation the magnitude: \(V=(\mathbf{a} \times\) b) - c corresponds to the volume of the parallelepiped
Given three vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{b}\) show, making use of the representation with the determinant that: \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}=\mathbf{a}
Demonstrate the property of BAC-CAB: \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}=\mathbf{b}(\mathbf{a} \cdot \mathbf{c})-\mathbf{c}(\mathbf{a} \cdot \mathbf{b})\), verifying it by direct
An automobile A travels on a straight road at the constant speed \(v_{A}=75 \mathrm{~km} / \mathrm{h}\). A second car B arrives at the speed \(v_{B}=130 \mathrm{~km} / \mathrm{h}\) and starts braking
A particle moves in uniformly accelerated motion on a straight line. After \(t_{1}=4 \mathrm{~s}\) it has traveled \(60 \mathrm{~m}\) and has a velocity \(v_{1}=33 \mathrm{~m} / \mathrm{s}\).
The equation of motion of a material point is expressed by the relation \(x(t)=\alpha t^{3}-\beta t^{2}-\gamma\), with the constants \(\alpha, \beta, \gamma\) real positive. Determine velocity and
Verify that the range \(x_{G}\) of a projectile is given by the (3.30). XG 2v sin cos 0 9 v sin 20 (3.30) 9
Two cannons are placed in the same position at different altitudes, \(h_{1}\) and \(h_{2}\). Two projectiles are fired simultaneously and horizontally. Calculate what ratio the two initial velocities
Verify that the launch angle \(\theta\) that produces the maximum range corresponds to \(45^{\circ}\). Verify further that in the absence of friction, although the trajectory changes, the range
A particle moves on a straight line with acceleration \(a(t)=\alpha t+\beta\), with \(\alpha=18 \mathrm{~m} / \mathrm{s}^{3}\) and \(\beta=-8 \mathrm{~m} / \mathrm{s}^{2}\). Calculate its velocity at
One wants to determine the depth \(h\) of a well experimentally. For this purpose, you throw a stone into it, and you hear the thud at the bottom after a time \(\tau=3.0 \mathrm{~s}\). The speed of
Determine the angular speed of the hour and minute hands of a clock (of course, with hands and not digital!). If at 3:00 (a.m. or p.m.) the angle between the hour hand and minute hand form an angle
An observer is stationary on a merry-go-round at a distance \(R=3 \mathrm{~m}\) from the axis of rotation. The merry-go-round, starting from a standstill, begins to rotate with constant angular
The position vector along a trajectory expressed in terms of the scalar distance \(s\) from the origin is given by the relation \(\mathbf{r}=\mathbf{a} s^{2}+\mathbf{b} s+\mathbf{c}\), with the
A particle is constrained to move on a circular guideway of radius \(R=3.00 \mathrm{~m}\), on which it can slide without friction, according to the motion equation law \(s(t)=k t^{3}\), with \(k=2.0
A particle moves on a predetermined trajectory with the equation of motion \(s(t)=k t^{2}\), with \(k\) constant and with magnitude of the acceleration equal to \(a=2 k\). Show by using (3.74) that
In the case considered in the question (14), show what the trajectory corresponds to in case the magnitude of acceleration is \(a=2 k \sqrt{1+\frac{t}{T}}\), where \(T=\) cost. Question 14A particle
The equation of motion of a particle is given in Cartesian coordinates by the relation \(\mathbf{r}(t)=\alpha t^{2} \hat{\mathbf{i}}+\beta t^{2} \hat{\mathbf{j}}\). Determine (i) the trajectory of
A projectile is launched from the Earth's surface with velocity \(v_{0}=50.0 \mathrm{~m} / \mathrm{s}\), at an angle \(\theta=60^{\circ}\) to the vertical. Determine the radius of curvature of the
The position of a particle is defined by the position vectorwhere numerically \(a=0.33, b=0.71, c=1.0\). Determine:1. the dimensions of \(a, b, c\);2. the function describing the velocity and
As an exercise in ballistics, we use the example of a soccer ball approximated as a point object (i.e., not subject to rotations, and the "effects" of the ball related to it). We also neglect all
A particle \(\mathrm{P}\) is moving on a circumference of radius \(R=3.50 \mathrm{~m}\) according to an angular speed given by \(\omega(t)=k t^{2}\), with \(k=13.18\) degrees \(/ \mathrm{s}^{3}\)
A particle is simultaneously subjected to a force of magnitude \(F_{1}=34 \mathrm{~N}\) along the negative direction of the \(x\) axis, and to a force of magnitude \(F_{2}=25 \mathrm{~N}\) that forms
A particle of mass \(m=650 \mathrm{~g}\), initially stationary, is subjected to the action of a constant force \(\mathbf{F}_{1}=34 \hat{\mathbf{i}} \mathrm{N}\). After a time \(t_{1}=1.5
A passenger car is traveling at \(v_{0}=11.1 \mathrm{~m} / \mathrm{s}\) downhill along a road of slope \(15^{\circ}\). At a certain instant it brakes by locking all wheels simultaneously until it
A body of mass \(m=2.4 \mathrm{~kg}\) slides on a rough horizontal plane, with static and dynamic coefficients of friction \(\mu_{s}=0.45\) and \(\mu_{d}=0.35\), respectively. If the body is
Two masses \(m_{1}=5.0 \mathrm{~kg}\) and \(m_{2}=2.0 \mathrm{~kg}\) are joined by an inextensible rope of negligible mass. The mass \(m_{1}\) rests on a rough horizontal plane while \(m_{2}\) is
An AMAZON pack of mass \(7.0 \mathrm{~kg}\) is stationary on a horizontal rough surface. The coefficient of static friction for the pack is \(\mu_{s}=0.85\). The pack sorting device acts with a force
Show that for the motion of a pendulum given by (4.40), the speed has maximum value in absolute value and negative sign at the time \(t=\pi / 2 \omega\). This causes the pendulum to continue motion
A stressed spring moves with harmonic motion of period \(T\). Its elongation at time \(t_{1}=T / 8\) \(\mathrm{s}\) is equal to \(x_{1}=2 \mathrm{~cm}\), and its velocity and acceleration are equal
A sphere of mass \(m=150 \mathrm{~g}\) rotates with constant angular frequency \(\omega=2.9 \mathrm{rad} / \mathrm{s}\) in a circular path on a horizontal plane, held on the circular trajectory by a
Two packs \(\mathrm{A}\) and \(\mathrm{B}\) of mass \(m_{A}=2.2 \mathrm{~kg}\) and \(m_{B}=2.8 \mathrm{~kg}\) are connected by an inextensible rope of negligible mass. Pack A rests on an inclined
A particle lies on a smooth plane inclined \(35^{\circ}\) with respect to the ground and resting on a spring of elastic constant \(k=55 \mathrm{~N} / \mathrm{m}\). The spring to support the body
A conical pendulum is a device analogous to a simple pendulum, except that the mass can rotate along a horizontal circumference in the \(x, y\) plane while the acceleration of gravity is along the
A box of mass \(M=38.6 \mathrm{~kg}\) rests on a horizontal plane with static friction coefficients \(\mu_{s}=0.810\) and dynamic friction coefficients \(\mu_{d}=0.525\). You want to move the box by
To a block of mass \(m=4.8 \mathrm{~kg}\) that is on an inclined plane at an angle \(\alpha=38^{\circ}\) to the horizontal, is applied the horizontal force, drawn in Fig. 4.13, of magnitude equal to
Consider a plane, inclined at an angle \(\alpha=20^{\circ}\) to the horizontal, rough, on which is placed a mass \(M=910 \mathrm{~g}\). The latter is connected to a bucket of mass \(m=490
Consider as in Fig. 4.14 a block of mass \(M=1.35 \mathrm{~kg}\) placed on a rough horizontal plane with dynamic coefficient of friction \(\mu_{d}=0.250\) to which a constant force \(\mathbf{F}\) is
A wedge of very large mass and angle \(\theta=30^{\circ}\) is resting on a horizontal surface as shown in Fig.4.15. Two bodies, of mass \(m_{1}=500 \mathrm{~g}\) and \(m_{2}\) are arranged as in the
Two blocks of mass \(M_{1}=1.20 \mathrm{~kg}\) and \(M_{2}=0.25 \mathrm{~kg}\) are hung as in Fig. 4.15: \(M_{1}\) is hung from the ceiling by an inextensible wire of negligible mass, while \(M_{2}\)
A solid spherical ball of radius \(R=1.0 \mathrm{~cm}\) and mass \(M=2.0 \mathrm{~g}\) rolls, without crawling, on a horizontal plane with center-of-mass velocity equal to \(v_{0}=0.80 \mathrm{~m} /
A train is leaving with constant acceleration of \(0.37 \mathrm{~m} / \mathrm{s}^{2}\). A ball is launched from the platform upward with initial velocity \(v_{0}=4.0 \mathrm{~m} / \mathrm{s}\).
A truck accelerates down a slope taking, starting at rest, \(6.00 \mathrm{~s}\) to reach a speed of \(30.0 \mathrm{~m} / \mathrm{s}\). An object of mass \(m=250 \mathrm{~g}\) is suspended via a rope

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