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If the latus rectum of an ellipse is equal to half of minor axis,then find its eccentricity.
answered Oct 9, 2014
Toolbox:Length of the latus rectum is $\large\frac{2b^2}{a}$Length of the minor axis is $2b$$e=\larg...
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Find the equation of a circle concentric with the circle $x^2+y^2-6x+12y+15=0$ and has double of its area.
answered Oct 9, 2014
Toolbox:The radius of the circle is $\sqrt{h^2+k^2-c}$ where $h$ and $k$ are the centres.Answer : $x...
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If the line $y=\sqrt 3x+k$ touches the circle $x^2+y^2=16$,then find the value of $k$.
answered Oct 8, 2014
Toolbox:The radius of a circle is always perpendicular to the tangent.The perpendicular distance fro...
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Find the equation of the circle having $(1,-2)$ as its centre and passing through $3x+y=14,2x+5y=18$.
answered Oct 8, 2014
Toolbox:General equation of the circle is $x^2+y^2+2hx+2ky+c=0$Answer : $x^2+y^2+2x-4y+20=0$Let the ...
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If one end of a diameter of the circle $x^2+y^2-4x-6y+11=0$ is $(3,4)$,then find the coordinate of the other end of the diameter.
answered Oct 8, 2014
Toolbox:The centre of a circle is the midpoint of the diameter.Answer : $(1,2)$Equation of the given...
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Find the equation of a circle which touches both the axes and the line $3x-4y+8=0$ and lies in the third quadrant.
answered Oct 8, 2014
Toolbox:The radius and a tangent to a circle are always perpendicular to each other.Length of the pe...
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If the lines $3x – 4y + 4 = 0$ and $6x – 8y – 7 = 0$ are tangents to a circle, then find the radius of the circle
answered Oct 8, 2014
Toolbox:If the slopes of two lines are equal,then the lines are parallel.The distance between the pa...
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A spring whose natural length is 15 cm exerts a force of 45 N when stretched to a length of 20 cm.Find the work done in stretching the spring from a length of 20 cm to a length of 25 cm
answered Sep 2, 2014
Answer : 3.375J The change of length is 5cm =.05m.Since $F=kx$ we have $45=k(.05)$ ...
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A force of 1200 N compresses a spring from its natural length of 18 cm to a length of 16 cm. How much work is done in compressing it from 16 cm to 14 cm?
answered Sep 2, 2014
Answer : 3600N.cmWe must work out the spring constant first (we will work in cm)$F=kx$So $1200 =k(2)...
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A spring is stretched 60 cm, and it takes a force of 40 Newtons to hold it there. How much work did it take to stretch the spring to this point?
answered Sep 2, 2014
Answer : 12 JoulesWe know that F = 40 Nwhen $x = 60 cm = 0.60$ m.Since $F = kx$$k = F/x = (40 N)/(0....
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A wall is tilted in by $15^{\large\circ}$,and a 2kg block is placed against the wall.The block is held in place with a spring.The coefficients of friction between the block and the wall are $\mu_s=0.61$ and $\mu_k=0.49$.Holding the block in place requires that the spring be compressed by 12cm.What is the spring constant of the spring?
answered Sep 2, 2014
Answer : 301N/mhttps://clay6.com/mpaimg/spring1.png$-N+kx-mg\sin 15^{\large\circ}+\mu_s N-mg\cos 15^...
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It takes 100J of work to stretch a spring 0.5m from its equilibrium position.How much work is needed to stretch it an additional 0.75m?
answered Sep 2, 2014
Answer : 525J$100 =\int_0^{0.5} kxdx$$\quad\;\;\;= \large\frac{1}{8}$$k$$\Rightarrow k=800$$W=\int\l...
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A force of 6 N is required to stretch a string from a natural length of 4 m to a length of 4.5 m.What work is required to stretch it to a length of 6 m.
answered Sep 2, 2014
Answer : 24JFirst we need to find the spring constant.Let $x$ denote the distance the spring is stre...
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Suppose 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm.Then how far beyond its natural length will a force of 30 N keep the spring stretched?
answered Sep 2, 2014
Answer : $\large\frac{27}{10}$From Hooke's law : $F=kx$ where $x$ is the distance the sprin is stret...
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Suppose 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm.Then how much work is needed to stretch it from 30 cm to 40 cm.
answered Sep 2, 2014
Answer : $\large\frac{1}{18}$From Hooke's law : $F=kx$ where $x$ is the distance the spring is stret...
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A spring has a length of 15.4 cm and is hanging vertically from a support point above it.A weight with a mass of 0.200 kg is attached to the spring,causing it to extend to a length of 28.6cm.What is the value of the spring constant?
answered Sep 1, 2014
Answer : 14.9N/m We place the origin of our coordinate system at the top of the spring,with the...
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A spring has a natural length of 18 inches and a force of 20 lbs is required to stretch and hold the spring to a length of 24 inches. What is the work required in ft-lbs to stretch the spring from a length of 21 inches to a length of 26 inches?
answered Sep 1, 2014
Answer : 7.6389ft-lbsWe are told that a force of 20lbs is needed to stretch the spring 24in-18in =6i...
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A spring has a natural length of 1 meter.A force of 25 Newtons stretches the spring by $\large\frac{1}{4}$ of a meter.Determine how much work is done by stretching the spring from a length of 1.5 meters to 2.5 meters.
answered Sep 1, 2014
Answer : 100JWe first determine the spring constant,$k$.Because the force is 25N when $x$ is 0.25m,w...
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A spring has natural length 30 cm,and a force of 50 Newtons(N) is required to extend it to 35 cm.How much work is required to extend it from 32 to 36 cm?
answered Sep 1, 2014
Answer : 160 N.cmWe have $F(35-30)=50N=k(35-30)cm\large\frac{N}{cm}$$=k (5cm)$So $k= \large\frac{50}...
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A spring has a natural length of 20 cm. A 40 N force is required to stretch (and hold the spring) to a length of 30 cm. How much work is done in stretching the spring from 35 cm to 38 cm?
answered Sep 1, 2014
Answer : 1.98JA force of 40N is required to stretch the spring $30cm-20cm=10cm=0.10m$ from its natur...
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A block whose mass $m$ is $15\; kg$ slides on a horizontal frictionless surface with a constant speed $v$ of $3.7m/s$.It is brought momentarily to rest as it compresses a spring in its path.By what distance $d$ is the spring compressed?The spring constant $k$ is $1500\;N/m$
answered Sep 1, 2014
Answer : 37 cmThe work done by the spring force on the block as the spring is compressed a distance ...
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A spring has a natural length of 14 meters.If a force of 5N is required to keep the spring stretched 2 meters.How much work is done in stretching the spring from its natural length to a length of 18 meters.
answered Sep 1, 2014
Answer : 20 JoulesWe must first find the spring constant $k$.By Hooke's law,$F(x) =kx$Thus,$F(2) =k(...
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A gas in a piston cylinder assembly undergoes an expansion process where $PV^{1.5}$=constant.The initial pressure is 3 bar,the initial volume is $0.1m^3$ and the final volume is $0.2m^3$.Determine the work done for this process.
answered Sep 1, 2014
Answer : 17.6kJhttps://clay6.com/mpaimg/workdone3.png$W_{1-2}=\int\limits_1^2 PdV$$\qquad\;\;=C\larg...
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We have a cable that weighs 2lbs/ft attached to a bucket filled with coal that weighs 800lbs.The bucket is initially at the bottom of a 500 ft mine shaft.Determine the amount of work required to lift the bucket to the midpoint of the shaft.
answered Sep 1, 2014
Answer : 387500 ft-lbAssume $x$ to be the amount of cable that has been pulled up.So at the bottom o...
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A particle of mass 0.5 kg moves on the positive $x$-axis under the action of a variable force $\large\frac{40}{x^2}$ newtons,directed away from origin..The particle passes through a point 2 metres from origin,with velocity $8ms^{-1}$ away from origin. Its experiences a constant resistance force of 6 newtons. Find the speed of the particle when it is 5 metres from origin.
answered Sep 1, 2014
Answer : $\sqrt{40}ms^{-1}$https://clay6.com/mpaimg/workdone2.pngThe work done by the resistance is ...
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Find the work done by F as it moves from a to b.
answered Sep 1, 2014
Answer : 625J $W=\int\limits_a^b\overrightarrow{F}. \overrightarrow{ds}$ $\;\;\;\;=\int...
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A 106g bullet is fired from a rifle having a barrel 0.441m long.Assuming the origin is placed where the bullet begins to move,the force exerted on the bullet by the expanding gas is $F=a+bx-cx^2$,where $a=13500N,b=8170N/m,c=24000N/m^2$,with $x$ in meters.With what velocity does the bullet exit the barrel?
answered Sep 1, 2014
Answer : 6061.83JThe work is found by integrating the force over the distance$W=\int\limits_{x_i}^{x...
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A 106g bullet is fired from a rifle having a barrel 0.271m long.Assuming the origin is placed where the bullet begins to move,the force exerted on the bullet by the expanding gas is F=a+bx-cx$^2$,where a=13500N, b=8170N/m,c=24000N/m$^2$,with x in meters.Determine the work done by the gas on the bullet as the bullet travels the length of the barrel.
answered Sep 1, 2014
Answer : 3799.29JThe work is found by integrating the force over the distance.$W=\int\limits_{x_i}^{...
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Force $F=(3x^2N)i+(4N)j$ acts on a particle changing only the kinetic energy of the particle.How much work is done on the particle as it moves from coordinates (2m,3m) to (3m,0m)?
answered Sep 1, 2014
Answer : 7.0J$W=\int\limits_2^3 3x^2dx+\int\limits_3^0 4dy$$\;\;\;\;=3\int\limits_2^3 x^2dx+4\int\li...
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What work is done by a force $F=(2xN)i+(3N)j$,with $x$ in meters,that moves a particle from a position $r_i=(2m)i+(3m)j$ to a position $r_f=-(4m)i-(3m)j$?
answered Sep 1, 2014
Answer : -6JWe use the general definition of work (for a two-dimensional problem),$W=\int\limits_{x_...
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A particle moves along the $x$-axis from $x=0$ to $x=1$ under the influence of the force $F=(1+x^2)^{-1}$.Find the work done by this force.
answered Sep 1, 2014
Answer : $\large\frac{\pi}{4}$The displacement variable is $x$ (not s) we can use the formula $dW = ...
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A force $F_x$ acts on a particle.The force is related to the position of the particle by the formula $F_x=Cx^3$,where $C$ is a constant.Find the work done by this force on the particle when the particle moves from $x=1.5\;m$ to $x=3\;m$.
answered Sep 1, 2014
Answer : 19C J$W=\int Fdx$$W=C\int\limits_{1.5}^3x^3dx$$\;\;\;\;=\large\frac{C}{4}$$(3^4-1.5^4)$$\;\...
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Work done in time t on a body of mass $m$ which is accelerated from rest to a speed $v$ in time $t_1$ as a function of time $t$ is given by
answered Sep 1, 2014
Answer : $\large\frac{1}{2}\normalsize m\large\frac{v^2}{t_1^2}\normalsize t^2$Work done =$F.s=ma\bi...
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The vessels A and B of equal volume and weight are immersed in water to a depth h.The vessel A has an opening at the bottom through which water can enter.If the work done in immersing A and B are $W_A$ and $W_B$ respectively,then
answered Aug 28, 2014
Answer : $W_A < W_B$When the vessels are immersed in water,work has to be done against up-thrust ...
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A particle is moved along the $x$-axis by a force which measures $\large\frac{10}{(1+x)^2}$ at a point $x$ feet from the origin.Find the work done (in ft-lb) in moving a particle from the origin to a distance of 9ft.
answered Aug 28, 2014
Answer : 9ft-lbW=$\int\limits_0^9\large\frac{10}{(1+x)^2}$$dx$$\;\;\;=-\large\frac{10}{1+x}\bigg|_0^...
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A particle moves under the effect of a force $F=Cx$ from $x=0$ to $x=x_1$.The work done in the process is
answered Aug 28, 2014
Answer : $\large\frac{1}{2}$$ Cx_1^2$Work done = $\int\limits_{x_1}^{x_2} Fdx $$\qquad\qquad=\int\li...
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A position dependent force $\overrightarrow{F}=(7-2x+3x^2)N$ acts on a small object of mass 2kg to displace it from $x=0$ to $x=5m$.The work done in joules is
answered Aug 28, 2014
Answer : 135 JWork done =$\int\limits_{x_1}^{x_2}Fdx$$\qquad\qquad=\int\limits_0^5(7-2x+3x^2)dx$$\qq...
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A cable weighing 2lbs/ft is used to lift 800 lbs of coal up a mineshaft 500 ft deep.Find out how much work is done in ft-lbs.
answered Aug 28, 2014
Answer : 650000 ft-lbs We divide into two parts , the work needed to lift the coal and the ...
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A man lifts 100 boxes each of mass 15 g to a height of 1.5 m. Find the work done by the man against gravity
answered Aug 28, 2014
Answer : 2250g JThe work done against gravity in lifting one box =$15g\times 1.5J=22.5g\;J$The work ...
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How much work is done when a constant force of 500 N is used to push a cart through a displacement of magnitude 25 m to final position along the $x$-axis
answered Aug 28, 2014
Answer : 12500 JAmount of Work is $W = F ⋅ d$$\qquad\qquad\qquad\qquad$=500 N.25 m$\qquad\qquad\qqua
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A constant force $F=(2\hat{i}+3\hat{j})$ newton displaces a body from position $r_1=(4\hat{i}-5\hat{j})$ metre to $r_2=(\hat{i}+3\hat{j})$ metre.Find the work done by the force.
answered Aug 28, 2014
Answer : 18JDisplacement $S=r_2-r_1$$\qquad\qquad\quad\;\;=(\hat{i}+3\hat{j})-(4\hat{i}-5\hat{j})$$\...
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How much work is done against gravity in lifting a 3kg object through a distance of 40cm?
answered Aug 28, 2014
Answer : 11.8JTo lift a 3kg object at constant speed,an upward force equal in magnitude to its weigh...
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A horizontal force F pulls a 20-kg carton across the floor at constant speed.If the coefficient of sliding friction between carton and floor is 0.60,how much work does F do in moving the carton 3.0m?
answered Aug 28, 2014
Answer : 353JBecause horizontal speed is constant,the carton is in horizontal equilibrium.$F = f=\mu...
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Figure shows the top view of two horizontal forces pulling a box along the floor the box is displaced 70cm along the broken line?What is the total work done by the two forces in pulling the box this distance?
answered Aug 28, 2014
Answer : 81.2JIn each case take the component of the force in the direction of the displacement :$(8...
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A horizontal force of 25N pulls a box along a table.How much work does it do in pulling the box 80cm?
answered Aug 28, 2014
Answer : 20JWork is force times displacement through which the force acts.Here,force is in the same ...
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A cyclist comes to a skidding stop in 10 m. During this process, the force on the cycle due to the road is 200 N and is directly opposed to the motion.How much work does the road do on the cycle?
answered Aug 28, 2014
Answer : -2000JWork done on the cycle by the road is the work done by the stopping (frictional) forc...
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A block of mass m = 2kg is raised vertically upwards by means of a massless string through a distance of S=4m with a constant acceleration a=2.2ms$^{-2}$.Find the net work done on the block.
answered Aug 28, 2014
Answer : 17.6JFrom the free body diagram https://clay6.com/mpaimg/work2.png$T-mg =ma$$\Rightarrow T=...
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A horizontal force F pulls a 20kg box at a constant velocity along a horizontal floor.If the coefficient of friction between the box and the floor is 0.25,find the work done by force F in moving the box through a distance of 2m.
answered Aug 28, 2014
Answer : 98JSince the box is moved at a constant velocity,the applied force F just overcomes the fri...
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A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F = 50.0 N at an angle of $30.0^{\large\circ}$ with the horizontal. Calculate the work done by the force on the vacuum cleaner as the vacuum cleaner is displaced 3 m to the right.
answered Aug 28, 2014
Answer : 130Jhttps://clay6.com/mpaimg/work1.pngThe work done by the force is defined to be the produ...
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It is well known that a raindrop falls under the influence of the downward gravitational force and the opposing resistive force. The latter is known to be proportional to the speed of the drop but is otherwise undetermined. Consider a drop of mass 1.00 g falling from a height 1.00 km. It hits the ground with a speed of $50.0 m s^{-1}$.What is the work done by the unknown resistive force?
answered Aug 28, 2014
Answer : -8.75JThe change in kinetic energy of the drop is$\Delta K = \large\frac{1}{2}$$mv^2$$\;\;\...
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