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Q. 118. A particle has shifted along some trajectory in the plane xy from point 1 whose radius vector r1 = i + 2j to point 2 with the radius vector r2 = 2i — 3j. During that time the particle experienced the action of certain forces, one of which being F = 3i + 4j. Find the work performed by the force F. (Here r1, r2, and F are given in SI units).
Ans. As is constant so the sought work done
Q. 119. A locomotive of mass m starts moving so that its velocity varies according to the law where a is a constant, and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.
Ans. Differentating v (s) with respect to time
(As locomotive is in unidrectional motion)
Hence force acting on the locomotive
Let, at v = 0 at t = 0 then the distance covered during the first t seconds
Hence the sought work,
Q. 120. The kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as T = as2, where a is a constant. Find the force acting on the particle as a function of S.
Ans. We have
Differentating Eq. (1) with respect to time
Hence net acceleration of the particle
Hence the sought force,
Q. 121. A body of mass m was slowly hauled up the hill (Fig. 1.29) by a force F which at each point was directed along a tangent to the trajectory. Find the work performed by this force, if the height of the hill is h, the length of its base l, and the coefficient of friction k.
Ans. Let makes an angle 0 with the horizontal at any instant of time (Fig.). Newton’s second law in projection form along the direction of the force, gives :
F = Jang cos θ + mg sin θ (because there is no acceleration of the body.)
Q. 122. A disc of mass m = 50 g slides with the zero initial velocity down an inclined plane set at an angle α = 30° to the horizontal; having traversed the distance l = 50 cm along the horizontal plane, the disc stops. Find the work performed by the friction forces over the whole distance, assuming the friction coefficient k = 0.15 for both inclined and horizontal planes.
Ans. Let 5 be the distance covered by the disc along the incline, from the Eq. of increment of M.E. of the disc in the field of gravity :
On puting the values Afr = -0.05 J
Q. 123. Two bars of masses m1 and m2 connected by a non-deformed light spring rest on a horizontal plane. The coefficient of friction between the bars and the surface is equal to k. What minimum constant force has to be applied in the horizontal direction to the bar of mass m1 in order to shift the other bar?
Ans. Let x be the compression in the spring when the bar m2 is about to shift Therefore at this moment spring force on m2 is equal to the limiting friction between the bar m2 and horizontal floor. Hence
k x - k m2g [where k is the spring constant (say)] (1)
For the block m1 from work-eneigy theorem : A - ΔT = 0 for minimum force. (A here indudes the work done in stretching the spring.)
From (1) and (2),
Q. 124. A chain of mass m = 0.80 kg and length l = 1.5 m rests on a rough-surfaced table so that one of its ends hangs over the edge. The chain starts sliding off the table all by itself provided the overhanging part equals η = 1/3 of the chain length. What will be the total work performed by the friction forces acting on the chain by the moment it slides completely off the table?
Ans. From the initial condition of the problem the limiting fricition between the chain lying on the horizontal table equals the weight of the over hanging part of the chain, i.e
(where λis the linear mass density of the chain)
Let (at an arbitrary moment of time) the length of the chain on the table is x. So the net friction force between the chain and the table, at this moment :
The differential work done by the friction forces :
(Note that here we have written ds = - dx., because ds is essentially a positive term and as the length of the chain decreases with time, dx is negative) Hence, the sought work done
Q. 125. A body of mass m is thrown at an angle α to the horizontal with the initial velocity v0. Find the mean power developed by gravity over the whole time of motion of the body, and the instantaneous power of gravity as a function of time.
Ans. The velocity of the body, t seconds after the begining of the motion becomes Thb power developed by the gravity at that moment, is (1)
As is a constant force, so the average power
where is the net displacement of the body during time of flight
Q. 126. A particle of mass m moves along a circle of radius R with a normal acceleration varying with time as wn = at2, where a is a constant. Find the time dependence of the power developed by all the forces acting on the particle, and the mean value of this power averaged over the first t seconds after the beginning of motion.
Ans. We have
t is defined to start from the begining of motion from rest
(where are unit vectors along the direction of tangent (velocity) and normal respectively)
Hence the sought average power
Q. 127. A small body of mass m is located on a horiiontal plane at the point O. The body acquires a horizontal velocity v0. Find:
(a) the mean power developed by the friction force during the whole time of motion, if the friction coefficient k = 0.27, m = 1.0 kg, and v0 = 1.5 m/s;
(b) the maximum instantaneous power developed by the friction force, if the friction coefficient varies as k = αx, where α is a constant, and x is the distance from the point O.
Ans. Let the body m acquire the horizontal velocity v0 along positive x - axis at the point O.
(a) Velocity of the body t seconds after the begining of the motion,
From Eq. (1), the time of motion τ - v0/kg
Hence sought average power during the time of motion
To find v (x), let us integrate the above equation
For maximum power,
Putting this value of x , in Eq. (2) we get,
Q. 128. A small body of mass m = 0.10 kg moves in the reference frame rotating about a stationary axis with a constant angular velocity ω = 5.0 rad/s. What work does the centrifugal force of inertia perform during the transfer of this body along an arbitrary path from point 1 to point 2 which are located at the distances r1 = 30 cm and r2 = 50 cm from the rotation axis?
Ans. Centrifugal force of inertia is directed outward along radial line, thus the sought work
Q. 129. A system consists of two springs connected in series and having the stiffness coefficients k1 and k2. Find the minimum work to be performed in order to stretch this system by Δl.
Ans. Since the springs are connected in series, the combination may be treated as a single spring of spring constant
From the equation of increment of M.E.,
Q. 130. A body of mass m is hauled from the Earth's surface by applying a force F varying with the height of ascent y as F = 2 (ay - 1) mg, where a is a positive constant. Find the work performed by this force and the increment of the body's potential energy in the gravitational field of the Earth over the first half of the ascent.
Ans. First, let us find the total height of ascent At the beginning and the end of the path of velocity of the body is equal to zero, and therefore the increment of the kinetic energy of the body is also equal to zero. On the other hand, in according with work-energy theorem ΔT is equal to the algebraic sum of the works A performed by all the forces, i.e. by the force F and gravity, over this path. However, since ΔT = 0 then A = 0. Taking into account that the upward direction is assumed to coincide with the positive direction of the y - axis, we can write
The work performed by the force F over the first half of the ascent is
The corresponding increment of the potential energy is
Q. 131. The potential energy of a particle in a certain field has the form U = a/r2 — blr, where a and b are positive constants, r is the distance from the centre of the field. Find:
(a) the value of r0 corresponding to the equilibrium position of the particle; examine whether this position is steady;
(b) the maximum magnitude of the attraction force; draw the plots U (r) and Fr (r) (the projections of the force on the radius vector r).
Ans. From the equation
(a) we have at r - r0, the particle is in equilibrium position.
To check, whether the position is steady (the position of stable equilibrium), we have to satisfy
Putting the value of
(as a and ft are positive constant)
which indicates that the potential energy of the system is minimum, hence this position is steady.
(b) We have
For F , to be maximum,
As Fr is negative, the force is attractive.
Q. 132. In a certain two-dimensional field of force the potential energy of a particle has the form U = αx2 + βy2, where α and β are positive constants whose magnitudes are different. Find out:
(a) whether this field is central;
(b) what is the shape of the equipotential surfaces and also of the surfaces for which the magnitude of the vector of force F = const.
Ans. (a) We have
For a central force,
Hence the force is not a central force.
According to the problem
Therefore the surfaces for which F is constant is an ellipse. For an equipotential surface U is constant.
Hence the equipotential surface is also an ellipse.
Q. 133. There are two stationary fields of force F = ayi and F = axi + byj, where i and j are the unit vectors of the x and y axes, and a and b are constants. Find out whether these fields are potential.
Ans. Let us calculate the workperformed by the forces of each field over the path from a certain point 1 (x1, y1) to another certain point 2 (x2 , y2)
In the first case, the integral depends on the function of type y (x), i.e. on the shape of the path. Consequently, the first field of force is not potential. In the second case, both the integrals do not depend on the shape of the path. They are defined only by the coordinate of the initial and final points of the path, therefore the second field of force is potential.
Q. 134. A body of mass in is pushed with the initial velocity v0 up an inclined plane set at an angle α to the horizontal. The friction coefficient is equal to k. What distance will the body cover before it stops and what work do the friction forces perform over this distance?
Ans. Let s be the sought distance, then from the equ ation of increm ent of M.E.
Q. 135. A small disc A slides down with initial velocity equal to zero from the top of a smooth hill of height H having a horizontal portion (Fig. 1.30). What must be the height of the horizontal portion h to ensure the maximum distance s covered by the disc? What is it equal to?
Ans. Velocity of the body at hight horizontally (from the figure given in the problem). Time taken in falling through the distance h.
(as initial vertical component of the velocity is zero.)
Putting this value of h in the expression obtained for s, we get,
smax = H
Q. 136. A small body A starts sliding from the height h down an inclined groove passing into a half-circle of radius h/2 (Fig. 1.31). Assuming the friction to be negligible, find the velocity of the body at the highest point of its trajectory (after breaking off the groove).
Ans. To complete a smooth vertical track of radius R, the minimum height at which a particle starts, must be equal to (one can proved it from energy conservation). Thus in our problem body could not reach the upper most point of the vertical track of radius R/2.
Let the particle A leave the track at some point O with speed v (Fig.). Now from energy conservation for the body A in the field of gravity :
From Newton s second law for the particle at the point O; Fn = mwn,
But, at the point O the normal reaction N = 0
A fter leaving the track at O, the particle A comes in air and further goes up and at maximum height of it's trajectory in air, it’s velocity (say v') becomes horizontal (Fig.). Hence, the sought velocity of A at this point
Q. 137. A ball of mass m is suspended by a thread of length l. With what minimum velocity has the point of suspension to be shifted in the horizontal direction for the ball to move along the circle about that point? What will be the tension of the thread at the moment it will be passing the horizontal position?
Ans. Let, the point of suspension be shifted with velocity vA in the horizontal direction towards left then in the rest frame of point of suspension the ball starts with same velocity horizontally towards right Let us work in this, frame. From Newton’s second law in projection form towards the point of suspension at the upper most point (say B) :
Condition required, to complete the vertical circle is that T > 0. But (2)
Again from eneigy conservation
From (4) and (5)
T = 3 mg
Q. 138. A horizontal plane supports a stationary vertical cylinder of radius R and a disc A attached to the cylinder by a horizontal thread AB of length l0 (Fig. 1.32, top view). An initial velocity v0 is imparted to the disc as shown in the figure. How long will it move along the plane until it strikes against the cylinder? The friction is assumed to be absent.
Ans. Since the tension is always perpendicular to the velocity vector, the work done by the tension force will be zero. Hence, according to the work energy theorem, the kinetic eneigy or velocity of the disc will remain constant during it’s motion. Hence, the sought time where s is the total distance traversed by the small disc during it's motion.
Now, at an arbitary position (Fig.)
It should be clearly understood that the only uncompensated force acting on the disc A in this case is the tension T, of the thread. It is easy to see that there is no point here, relative to which the moment of force T is invarible in the process of motion. Hence conservation of angular momentum is not applicable here.
Q. 139. A smooth rubber cord of length l whose coefficient of elasticity is k is suspended by one end from the point O (Fig. 1.33). The other end is fitted with a catch B. A small sleeve A of mass m starts falling from the point O. Neglecting the masses of the thread and the catch, find the maximum elongation of the cord.
Ans. Suppose that Δl is the elongation of the rubbler cord. Then from eneigy conservation,
Since the value of is certainly greater than 1, hence negative sign is a voided.
Q. 140. A small bar A resting on a smooth horizontal plane is attached by threads to a point P (Fig. 1.34) and, by means of a weightless pulley, to a weight B possessing the same mass as the bar itself. Besides, the bar is also attached to a point 0 by means of a light nondeformed spring of length l0 = 50 cm and stiffness x = 5 mg/l0, where m is the mass of the bar. The thread PA having been burned, the bar starts moving. Find its velocity at the moment when it is breaking off the plane.
Ans. When the thread PA is burnt, obviously the speed of the bars will be equal at any instant of time until it breaks oft. Let v be the speed of each block and θ be the angle, which the elongated spring makes with the vertical at the moment, when the bar A breaks off the plane. At this stage the elongation in the spring.
Since the problem is concerned with position and there are no forces other than conservative forces, the mechanical energy of the system (both bars + spring) in the field of gravity is conserved, i.e. ΔT + ΔU = 0
From Newton’s second law in projection form along vertical direction :
Taking simultaneous solutiorisof (2) and (3) yields :