Questions and Answers of Classical Dynamics Of Particles

For the geared system shown in Figure, assume that shaft inertias and the gear inertias l1, l2, and I3, are negligible. The motor and load inertias in kg-m2 areI4 = 0.03 I5 = 0.15The speed
For the geared system discussed in Problem 3.23, shown in Figure, the inertias are given in kg-m2 asI1 = 10-3I2 = 3.84 x 10-3I3 = 0.0148I4 = 0.03I5 = 0.15The speed ratios areÏ‰1/
The geared system shown in Figure P3.25 is similar to that used in some vehicle transmissions. The speed ratios (which arc the ratios of the gearradii) are Ï‰2/Ï‰1 =
Consider the rack-and-pinion gear shown in Figure. Use the free-body diagram method to obtain the expression for the acceleration in terms of the given quantities: R, T, m, and I.
For the conveyor system shown in Figure, the reducer reduces the motor speed by a factor of 2:1. The motor inertia is I1 = 0.003 kg-m2. Discount the inertias of the reducer and the tachometer, which
The lead screw (also called a power screw or a jack screw) is used to convert the rotation of a motor shaft into a translational motion of the mass m (see Figure). For one revolution of the screw,
At time t = 0, the operator of the road roller disengages the transmission so that the vehicle rolls down the incline (see Figure). Determine an expression for the vehicle's speed as a function of
For the mass shown in Figure 3.1.3b, m = 10 kg, ( = 25°, v(0) = 2 m/s, and μ = 0.3. Determine whether the mass comes to rest if (a) ft = 100 N and (b) f1 = 50 N. If the mass comes to rest, compute
Derive the equation of motion of the block of mass m1 in terms of its displacement x (see Figure). The friction between the block and the surface is negligible. The pulley has negligible inertia and
Assume the cylinder in Figure rolls without slipping. Neglect the mass of the pulleys and derive the equation of motion of the system in terms of the displacement x.
A conveyor drive system to produce translation of the load is shown in Figure. The reducer is a gear pair that reduces the motor speed by a factor of 10:1. The motor inertia is I = 0.002 kg m2. The
A person pushes a roller of radius R and inertia mR2/2, with a force f applied at an angle of cp to the horizontal (see Figure). The roller weighs 800 N and has a diameter of 0.4 m. Assume the roller
A slender rod l .4 m long and of mass 20 kg is attached to a wheel of radius 0.05 m and negligible mass, as shown in Figure. A horizontal force f is applied to the wheel axle. Derive the equation of
A slender rod l .4 m long and of mass 20 kg is attached to a wheel of mass 3 kg and radius 0.05 m, as shown in Figure. A horizontal force / is applied to the wheel axle. Derive the equation of motion
Consider the rolling cylinder treated in Example 3.4.1 and shown in Figure. Assume now that the no-slip condition is not satisfied, so that the cylinder slips while it rolls. Derive expression for
A hoop of mass m and radius r starts from rest and rolls down an incline at an angle θ. The hoop's inertia is given by IG = mr2. The static friction coefficient is μs. Determine the acceleration of
The pendulum shown in Figure consists of a slender rod weighing 3 lb and a block weighing 10 lb.a. Determine the location of the center of mass.b. Derive the equation of motion in terms of
The scale shown in Figure measures the weight mg of an object placed on the scale, by using a counterweight of mass mc. Friction in the pivot point at A causes the pointer to eventually come to rest
A particle of mass m = 19 kg slides down a frictionless ramp starting from rest Suppose that 0 = 30°, L = 5 m, and H = 2 m. Compute the distance D of the impact point.
A single link of a robot arm is shown in Figure. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by amotor torque Tm through two pairs of spur gears.
It is required to determine the maximum acceleration of the rear-wheel-drive vehicle shown in Figure. The vehicle mass is 1700 kg, and its dimensions are LA = 1.2 m, LB = 1.1m, and H = 0.5m. Assume
Figure illustrates a pendulum with a base that moves horizontally. This is a simple model of an overhead crane carrying a suspended load with cables. The load mass is m, the cable length is L, and
Figure illustrates a pendulum with a base that moves. The base acceleration is a(t). Derive the equation of motion in terms of Î¸ with a(t) as the input. Neglect the mass of the rod.
The overhead trolley shown in Figure is used to transport beams in a factory. The beam is rectangular, with a length of L. It is desired to limit the trolley horizontal acceleration a so that the
The analysis of the personal transporter in Example 3.5.6 assumed that the driving force / was the given input. Instead, model the system with the assumption that the input is the total torque Tw
The "sky crane" shown on the text cover was a novel solution to the problem of landing the 2000 lb Curiosity rover on the surface of Mars. Curiosity hangs from the descent stage by 60-foot-long nylon
A particle of mass m slides down a frictionless ramp starting from rest. The lengths L and H and the angle 0 are given. Derive an expression for the distance D of the impact point.
Radar tracks the flight of a projectile. At time the radar measures the horizontal component vx(t) and the vertical component vy(t) of the projectile's velocity and its range R(t) and elevation ((t).
A motor supplies a moment M to the pulley of radius R1. A belt connects this pulley to pulley A. Pulleys A and B form a rigid body with a common hub. Ignore the inertias of the three pulleys.
Compute the translational spring constant of a particular steel helical coil spring, of the type used in automotive suspensions. The coil has six turns. The coil diameter is 4 in., and the wire
Compute the equivalent tensional spring constant of the stepped shaft arrangement shown in Figure. For the shaft material, G = 8 x 1010 N/m2.
Plot the spring force felt by the mass shown in Figure as a function of the displacement x. When x = 0, spring 1 is at its free length. Spring 2 is at its free length in the configuration shown.
Calculate the expression for the natural frequency of the system shown in Figure. Disregard the pulley mass.
Obtain the expression for the natural frequency of the system shown in Figure. Assume small motions and disregard the pulley mass.
Obtain the expression for the natural frequency of the system shown in Figure. Discount the mass of the L-shaped arm.
A connecting rod having a mass of 3.6 kg is shown in Figure. It oscillates with a frequency of 40 cycles per minute when supported on a knife edge, as shown. Its mass center is located 0.15 m below
Calculate the expression for the natural frequency of the system shown in Figure.
For each of the systems shown in Figure, the input is the force f and the outputs are the displacements x1 and x2 of the masses. The equilibrium positions with f = 0 correspond to x1 = x2 = 0.
The mass m in Figure is attached to a rigid lever having negligible mass and negligible friction in the pivot. The input is the displacement x. When x and Î¸ are zero, the springs are at
In the pulley system shown in Figure, the input is the applied force f, and the output is the displacement x. Assume the pulley masses are negligible and derive the equation of motion.
In the spring arrangement shown in Figure, the displacement x is caused by the applied force f. Assuming the system is in static equilibrium, sketch the plot of f versus x. Determine the equivalent
Figure illustrates a cylindrical buoy floating in water with a mass density (. Assume that the center of mass of the buoy is deep enough so that the buoy motion is primarily vertical. The buoy mass
Figure shows the cross-sectional view of a ship undergoing rolling motion. Archimedes' principle states that the buoyancy force B acting on a floating object equals the weight of the liquid displaced
In the system shown in Figure, the input is the angular displacement ( of the end of the shaft, and the output is the angular displacement Î¸ of the inertia I. The shafts have tensional stiffness's
In Figure, assume that the cylinder rolls without slipping. The spring is at its free length when x and y are zero,(a) Derive the equation of motion in terms of x, with y(t) as the input,(b) Suppose
In Figure when x1 = x2 = 0 the springs are at their free lengths. Derive the equations of motion.
In Figure model the three shafts as mass less tensional springs. When Î¸1= Î¸2 = 0 the springs are at their free lengths. Derive the equations of motion with the torque T2 as the input.
In Figure when Î¸1 = Î¸2 = 0 the spring is at its free length. Derive the equations of motion, assuming small angles.
Consider the torsion-bar suspension shown in Figure 4.1.6. Assume that the torsion bar is a steel rod with a length of 4 ft and diameter 1.5 in. The wheel weighs 40 lb and the suspension arm is 2 ft
For the system shown in Figure, suppose that k1 = k, k2 = k3 = 2k, and m1 = m2 = m. Obtain the equations of motion in terms of x1 and x2.
For the system shown in Figure, suppose that R2 = 2R1, m1 = m, and m2 = 2m. The two pulleys share a common hub and are welded together. Their total mass is m2 and total inertia is l2. Obtain the
In the arrangement shown in Figure, a cable is attached to the end of a cantilever beam. We will model the cable as a rod. Denote the translational spring constant of the beam by kb, and the
For Figure, assume that the cylinder rolls without slipping and use conservation of energy to derive the equation of motion in terms of x.
For Figure, the equilibrium position corresponds to x = 0. Neglect the masses of the pulleys and assume that the cable is inextensible, and use conservation of energy to derive the equation of motion
For Figure, the equilibrium position corresponds to x = 0. Neglect the masses of the pulleys and assume that the cable is inextensible, and use conservation of energy to derive the equation of motion
Use the Rayleigh method to obtain an expression for the natural frequency of the system shown in Figure. The equilibrium position corresponds to x = 0.
For Figure, assume that the cylinder rolls without slipping and use the Rayleigh method to obtain an expression for the natural frequency of the system. The equilibrium position corresponds to x = 0.
Use the Rayleigh method to obtain an expression for the natural frequency of the system shown in Figure. The equilibrium position corresponds to x = 0.
Use an energy method to obtain the expression for the natural frequency of the system shown in Figure.
Determine the natural frequency of the system shown in Figure using Rayleigh's method. Assume small angles of oscillation.
Determine the natural frequency of the system shown in Figure using an energy method. The disk is a solid cylinder. Assume small angles of oscillation.
Use Rayleigh's method to calculate the expression for the natural frequency of the system shown in Figure. Assume small motions and neglect the pulley mass.
In the spring arrangement shown in Figure, the displacement x is caused by the applied force f. Assuming the system is in static equilibrium when x = 0 and that the angle Î¸ is small,
Use Rayleigh's method to obtain the expression for the natural frequency of the system shown in Figure. Disregard the mass of the L-shaped arm.
Determine the natural frequency of the system shown in Figure using Rayleigh's method. Assume small angles of oscillation.
Figure shows an engine valve driven by an overhead camshaft. The rocker arm pivots about the fixed point O and the inertia of the arm about this point is Ir. The valve mass is mv and the spring mass
The vibration of a motor mounted on the end of a cantilever beam can be modeled as a mass-spring system. The motor weighs 30 lb, and the beam weighs 7 lb. When the motor is placed on the beam, it
The vibration of a motor mounted in the middle of a fixed-end beam can be modeled as a mass-spring system. The motor mass is 40 kg, and the beam mass is 13 kg. When the motor is placed on the beam,
The vibration of a motor mounted in the middle of a simply-supported beam can be modeled as a mass-spring system. The motor mass is 30 kg, and the beam mass is 10 kg. When the motor is placed on the
A certain cantilever beam vibrates at a frequency of 5 Hz when a 30 lb motor is placed on the beam. The beam weighs 7 lb. Estimate the beam stiffness k.
A 10-kg mass is attached to a 2 kg spring. The mass vibrates at a frequency of 20 Hz when disturbed. Estimate the spring stiffness k.
The static deflection of a cantilever beam is described byxy = P/(6EIA) y2 (3L - y)Where P is the load applied at the end of the beam, and xy is the vertical deflection at a point a distance y from
Figure shows a winch supported by a cantilever beam at the stern of a ship. The mass of the winch is mw, the mass of the beam plus winch bracket and motor is mb. The object hoisted by the winch has a
For the system shown in Figure, assume that the resulting motion is small enough to be only horizontal, and determine the expression for the equivalent ke that relates the applied force f to the
A 50-kg block is placed on an inclined plane whose angle with the horizontal is 25°. The viscous friction coefficient between the block and the plane is c = 6 N s/m.(a) Derive the equation of
A certain mass-spring-damper system has the following equation of motion.Suppose that the initial conditions are zero and that the applied force f(t) is a step function of magnitude 5000. Solve for
For each of the systems shown in Figure, the input is the force f and the outputs are the displacements x1 and x2 of the masses. The equilibrium positions with f = 0 correspond to x1 = x2 = 0.
In Figure a motor supplies a torque T to turn a drum of radius R and inertia I about its axis of rotation. The rotating drum lifts a mass m by means of a cable that wraps around the drum. The drum's
Derive the equation of motion for the lever system shown in Figure P4.54, with the force f as the input and the angle Î¸ as the output. The position Î¸ = 0 corresponds to the
In the system shown in Figure, the input is the displacement y and the output is the displacement x of the mass m. The equilibrium position corresponds to x = y = 0. Neglect any friction between the
Figure a shows a Houdaille damper, which is a device attached to an engine crankshaft to reduce vibrations. The damper has an inertia Id that is free to rotate within an enclosure filled with viscous
Refer to Figure. Determine the relations between c, c1, and c2 so that the damper shown in part (c) is equivalent to (a) the arrangement shown in part (a), and the arrangement shown in part (b).a.b.
For the system shown in Figure, obtain the equation of motion in terms of Î¸. The disk is a solid cylinder. Assume small angles of oscillation.
Find the transfer function Z(s)X(s) for the system shown in Figure.
The two stepped solid cylinders in Figure consist of the same material and have an axial force f applied to them. Determine the equivalent translational spring constant for this arrangement.
Find the transfer function Y(s)X(s) for the system shown in Figure.
Find the transfer function Y(s)X(s) for the system shown in Figure.
The mass m in Figure is attached to a rigid rod having inertia I about the pivot and negligible pivot friction. The input is the displacement z. When z = Î¸ = 0, the spring is at its free
In the system shown in Figure, the input is the force f and the output is the displacement xA of point A. When x = xA the spring is at its free length. Derive the equation of motion.
In the system shown in Figure, the input is the displacement y and the output is the displacement x. When x = y = 0 the springs are at their free lengths. Derive the equation of motion.
Figure shows a rack-and-pinion gear in which a damping force and a spring force act against the rack. Develop the equivalent rotational model of the system with the applied torque T as the input
Figure shows a drive train with a spur-gear pair. The first shaft turns N times faster than the second shaft. Develop a model of the system including the elasticity of the second shaft. Assume the
Assuming that Î¸ is small, derive the equations of motion of the systems shown in parts (a) and (b) of Figure. When Î¸ = 0 the systems are in equilibrium. Are the systems
Assuming that Î¸ is small, derive the equation of motion of the pendulum shown in Figure. The pendulum is in equilibrium when Î¸ = 0. Is the system stable, neutrally stable,
Assuming that Î¸ is small, derive the equation of motion of the pendulum shown in Figure. The input is y(t) and the output is Î¸. The equilibrium corresponds to y =
A table with four identical legs supports a vertical force. The solid cylindrical legs are made of metal with E = 2 x 1011 N/m2. The legs are 1 m in length and 0.03 m in diameter. Compute the
Figure shows a quarter-car model that includes the mass of the seats (including passengers). The constants k3, and c3 represent the stiffness and damping in the seat supports. Derive the equations of
The top view of a solid door is shown in Figure. The door has a mass of 40 kg and is 2.1 m high, 1.2 m wide and 0.05 m thick. Its door closer has a tensional spring constant of 13.6 N-m/rad. The door
Derive the equation of motion for the system shown in Figure. Assume small angles of oscillation, and neglect the rod mass. What relation among L1, L2, m, and k must be satisfied for the system to be
A boxcar moving at 1.3 m/s hits the shock absorber at the end of the track (Figure). The boxcar mass is 18 000 kg;. The stiffness of the absorber is k = 73 000 N/m, and the damping coefficient is c =

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