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Classical Mechanics

Problem 1

You may need to modify the Newtonian theory of gravitation at short ranges. Suppose the potential energy between 2 masses m and m' is:

V(r) = \frac{Gmm'}{r}=(1-ae^{-r / \pi})

For short distances r << π, calculate the force between m and m'

Solution 1

Recall the Newtonian theory of gravitation

F = G \frac{mM}{r^2} = \frac{mv^2}{r}

In this case, M = m' and you only need to remember the middle part

G \frac{m m'}{r^2}

Problem 2

A musicall instrument called the "bugle" is dropped into a wishing well. t is the time between releasing the bugle and hearing it clang on the bottom. If the speed of sound is 330 m/s and t = 2.059s what is the depth h of the well?

Solution 2

d = \frac{1}{2}gt^2
d = 19.6 \mathrm m

Problem 3

Calculate the centripetal force required to keep a 4 kg mass moving in a horizontal circle of radius 0.8 m at a speed of 6 m/s. (r is the radical vector with respect to the center)

Solution 3

Centripetal force is toward the center of the circle.

F = \frac{mv^2}{r}
F = \frac{(4) (6)^2}{0.8} = 180 N
F = -180 N \textbf{r}

Problem 4

A spring-mass system is suspected from the ceiling of an elevator. When the elevator is at rest, the system oscillates with period T. The elevateor now starts and moves upward with acceleration of a = 0.2 g. During this constant acceleration phase, what is the period T of the spring-mass system?

Solution 4

The spring mass system is independent of gravity, and given by

T = 2 \pi \sqrt{m/k}

Hence, the acceleration of the elevator will not affect the period.

Problem 5


Each of the figures above shows blocks of mass 2m and m acted on by an external horizontal force F. For each figure, which of the following statements about the magnitude of the force that one block exerts on the other (F12) is correct? (Assume that the surface on which the blocks move is frictionless.)

Solution 5


Problem 6

A simple pendulum of length l is suspended fro the ceiling of an elevator that is accelerating upward with constant acceleration a. For small oscillations, what is the period T of the pendulum?

Solution 6


Problem 7


A uniform disk with a mass of m and a radius of r rolls without slipping along a horizontal surface and ramp. The disk has an initial velocity of u. What is the maximum height h to which the center of mass of the disk rises?

Solution 7


Problem 8


A mass m attached to the end of a massless rod of length L is free to swing below the plane of support, as shown in the figure above. The Hamiltonian for this system is given by

H = \frac{p_{\theta}^2}{2 m L^2} + \frac{p_{\phi}^2}{2 m L^2 \mathrm{sin}^2 \theta} - mgL \mathrm{cos} \theta

where q and f are defined as shown in the figure. On the basis of Hamilton's equations of motion, the generalized coordinate or momentum that is a constant in time is

Solution 8


Problem 9


A rod of length L and mass M is placed along the x-axis with one end at the origin, as shown in the figure above. The rod has linear mass density

\lambda = \frac{2M}{L^2}x

where x is the distance from the origin. What is the x-coordinate of the rod's center of mass in relation to L?

Solution 9


Problem 10

Which of the following best illustrates the acceleration of a pendulumn bob at points a through e?

  1. 601
  2. 601
  3. 601
  4. 601
  5. 601

Solution 10


Problem 11


A stone is thrown at an angle of 45° above the horizontal x-axis in the +x-direction. Ignore air resistance. Which of the velocity versus time graphs shown above best represents vx versus t and vy versus t, respectively?

  1. I, IV
  2. II, I
  3. II, III
  4. II, V
  5. IV, V

Solution 11

If a stone is thrown at such an angle at an initial velocity, its horizontal vx vs t graph should be constant and positive, like Graph II


Thus, choices (A) and (E) are out.

Recall the basic kinematics equation


Eliminate choice (D), since that shows a parabolic time dependence, when a linear one is required. Since the slope is negative, the vy-graph should look like III. This leaves only one answer: (C)

If you forget the basic equations above, you can derive it all from summing up the net force

\ddot{y} = -g

Integrate both sides to get velocity. Integrate again to get position.

Problem 12


Seven pennies are arranged in a hexagonal planar pattern so as to touch each neighbor as shown in the figure above. Each penny is a uniform disk of mass m and radius r. What is the moment of inertia of the system of seven pennies above an axis that passes through the center of the central penny and is normal to the plane of the pennies?

  1. (7/2) m r2
  2. (13/2) m r2
  3. (29/2) m r2
  4. (49/2) m r2
  5. (55/2) m r2

Solution 12


Problem 13


A thin uniform rod of mass M and length L is positioin vertically aboeve an anchored frictionless pivot point, and then allowed to fall to the ground. With what speed does the free end of the rod strike the ground?

  1. √(g L / 3)
  2. √(g L)
  3. √(3 g L)
  4. √(12 g L)
  5. 12 √(g L)

Solution 13


Problem 14


The figure above shows a plot of the time-dependent force Fx(t) acting on a particle in motion along the x-axis. What is the total impulse delivered to the particle?

  1. 0
  2. 1 kg m / s
  3. 2 kg m / s
  4. 3 kg m / s
  5. 4 kg m / s

Solution 14


Problem 15


A particle of mass m is moving along the x-axis with speed v when it collides with a particle of mass 2m initially at rest. After the collision the first particle has come to rest and the second particle has split into two equal-mass pieces that move at equal angles θ > 0 with the x-axis. Which of the following statements correcty describes the speeds of the 2 pieces?

  1. Each piece moves with speed greater than v/2
  2. Each piece moves with speed v/2
  3. One of the pieces moves with speed v, the other moves with speed less than v.
  4. Each piece moves with speed v
  5. One of the pieces moves with speed v/2, the other moves with speed greater than v/2

Solution 15


Problem 16


Two identical blocks are connected by a spring. The combination is suspended at rest from a string attached to the ceiling as shown in the figure above. The string breakes suddenly. Immediately after the string breakes what is the downward acceleration of the upper block?

  1. √(2) g
  2. g / 2
  3. g
  4. 0
  5. 2 g

Solution 16


Problem 17


The cylinder shown above, with mass M and radius R, has a radially dependent density. The cylinder starts from rest and rolls without slipping down an inclined plane of height H. At the bottom of the plane its translational speed is (8 g H / 7)1/2. What is the rotational inertia of the cylinder?

Solution 17


Problem 18


Two small equal masses m are connected by an ideal massless spring that has equilibrium length ℓ0 and force constant k. The system is free to move without friction in the plane of the page. If p1 and p2 represent the magnitudes of the momenta of the 2 masses, what is a Hamiltonian for this system?

Solution 18


Problem 19


A 2kg box hangs by a massless rope from a ceiling. A force slowly pulls the box horizontally to the side until the horizontal force is 10N. The box is then in equilibrium as shown above. The angle that the rope makes with the vertical is closes to:

  1. arcsin 0.5
  2. arctan 0.5
  3. arctan 2.0
  4. 45 degrees
  5. arcsin 2.0

Solution 19

F_x = 0 = T \mathrm{sin}\theta - F
F_y = 0 = T \mathrm{cos}\theta - mg
\frac{F_x}{F_y} = \frac{T \mathrm{sin}\theta}{T \mathrm{cos}\theta}
\mathrm{tan} \theta = \frac{F}{mg} = \frac{10}{2 \cdot 10} = \frac{1}{2}
\mathrm{tan} \theta = 0.5
\theta = \mathrm{arctan}0.5
The answer is B

Problem 20


3 masses are connected by 2 springs as shown above. A longitudinal normal mode with frquency (below) is exhibited by what motion of A, B, and C?

\frac{1}{2 \pi} \sqrt{\frac{k}{m}}

Solution 20


Problem 21


Two wedges, each of mass m, are placed next to each other on a flat floor. A cube of mass M is balanced on the wedges as shown above. Assume no friction between the cube and the wedges, but a coefficient of static friction μ < 1 between the wedges and the floor. What is the largest M that can be balanced as shown without motion of the wedges?

  1. \frac {\mu m}{\sqrt{2}}
  2. \frac {\mu m}{1 - \mu}
  3. \frac {m}{\sqrt{2}}
  4. \frac {2 \mu m}{1 - \mu}
  5. All M will balance

Solution 21


Problem 22


A cylindrical tube of mass M can slide on a horizontal wire. Two identical pendulums, each of mass m and length l, hang from the ends of the tube, as shown above. For small oscillations of the pendulums in the planet of the paper, the eigenfrequencies of the normal modes of oscillation of this system are 0,



  1. \sqrt {\frac{g (M + m)}{l m}}
  2. \sqrt {\frac{g m}{l(M + 2m)}}
  3. \sqrt {\frac{g}{m}}
  4. \sqrt {\frac{g m}{l(M + m)}}
  5. \sqrt {\frac{g m}{l M}}

Solution 22


Problem 23


A solid cone hangs from a frictionless pivot at the origin O, as shown above. If î, ĵ, k̂ are unit vectors, and a, b, and c are positive constants, which of the following forces F applied to the rim of the cone at a point P results in a torque τ on the cone with a negative component τz?

  1. F = ak̂, P is (0, b, -c)
  2. F = aĵ, P is (b , 0, -c)
  3. F =-ak̂, P is (0, -b, -c)
  4. F =-ak̂, P is (-b, 0, -c)
  5. F = aĵ, P is (-b, 0, -c)

Solution 23


Problem 24


A car travels with constant speed on a circular road on level ground. In the diagram above, Fair is the force of air resistance on the car. Which of the other forces shown best represents the horizontal force of the road on the car's tires?

  1. FA
  2. FB
  3. FC
  4. FD
  5. FE

Solution 24


Problem 25


A block of mass m sliding down an incline at constant speed is initially at heigh h above the ground, as shown in the figure above. The coefficient of kinetic friction between the mass and the include is μ. If the mass continues to slide down the incline at a constant speed, how much energy is dissipated by friction by the time the mass reasches the bottom of the incline?

  1. μ m g h / sinθ
  2. 0
  3. m g h sinθ
  4. m g h / μ
  5. m g h

Solution 25


Problem 26


Three equal masses m are rigidly connected to each other by massless rods of length ℓ forming an equilateral triangle. The assembly is to be given an angular velocity ω about an axis perpendicular to the triangle. For fixed ω the ratio of the kinetic energy of the assembly for an axis through B compared with that for an axis through A is equal to:

  1. 3
  2. 2
  3. 1
  4. 1/2
  5. 1/3

Solution 26


Problem 27


A bead is constrained to slide on a frictionless rod that is fixed at an angle θ with a vertical axis and is rotating with angular frequency ω about the axis. Taking the distance s along the rod as the variable, what is the Lagrangian for the bead?

Solution 27


Problem 28


A string consists of 2 parts attached at x = 0. The right part of the string (x > 0) has a mass μr per unit length and the left part of the string (x < 0) has mass μ per unit length. The string tension is T. If a wave of unit amplitude travels along the left part of the string, what is the amplitude of the wave that is transmitted to the right part of the string?

Solution 28


Problem 29


Consider a particle moving without friction on a rippled surface, as shown above. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos(kx). If the particle starts at x=0 with a speed v in the x direction for what values of v will the particle stay on the surface at all times?

Solution 29


Problem 30


Two pendulums are attached to a massless spring, as shown above. The arms of the pendulums are of identical lengths ℓ, but the pendulum balls have unequal masses m1and m2. The initial distance between the masses is the equilibrium length of the spring, which has spring constant K. What is the highest normal mode frequency of the system?

Solution 30

\sqrt{\frac{g}{l} + \frac{K}{m_1} + \frac{K}{m_2}}

Problem 31


Small amplitude standing waves of wavelength λ occur on a string with tension T, mass per unit length μ and length L. One end of the string is fixed and the other end is attached to a ring of mass M that slides on a frictionless rod, as shown in the figure above. When gravity is neglected, which of the following conditions correctly determines the wavelength? (You might want to consider the limiting cases M goes to 0 and M goes to infinity)

\frac{\mu}{M} = \frac{2 \pi}{\lambda} \mathrm{cot} \frac{2 \pi}{\lambda}
\frac{\mu}{M} = \frac{2 \pi}{\lambda} \mathrm{tan} \frac{2 \pi}{\lambda}
\frac{\mu}{M} = \frac{2 \pi}{\lambda} \mathrm{sin} \frac{2 \pi}{\lambda}
\lambda = \frac{2L}{ n}, n = 1, 2, 3,...
\lambda = \frac{2L}{ n + 1/2}, n = 1, 2, 3,...

Solution 31