Classical Mechanics Answers

4. A student chased by the bull runs through the gate with uniform mass of 153 Kg. The gate is 2.8 m wide and 1.5 m high. The bull will escape if the gate is not closed in 30 sec. The student pushes the gate with 100N force directed perpendicular to the line at its edge 2.8 m from the hinge. a. Calculate the moment of inertia of the gate. b. Angular acceleration of the gate c. Time needed to close the gate if it starts at rest and rotates through 50 before closing.

5. A 15-kg object and a 10-kg object are suspended, joined by a cord that passes over a pulley with a radius of 15 cm and a mass of 3 kg. The cord has a negligible mass and does not slip on the pulley. Treat the pulley as a uniform disk, and determine the linear acceleration of the two objects after the objects are released from rest.

1. The tub of a washer goes into its spin cycle, starting from rest and gaining angular speed steadily for 8 s, at which time it is turning at 5 rev/s. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub smoothly slows to rest in 12 s. Through how many revolutions does the tub turn while it is in motion?

2. A solid sphere rolls without slipping along a horizontal surface. What percentage of its total kinetic energy is rotational kinetic energy?

3. Use the parallel axis theorem to find the moment of inertia of a) solid cylinder about an axis parallel to the center of mass and passing through the edge of the cylinder and b) a hollow sphere about an axis tangent to the surface of the sphere.

If the simple pendulum is brought from one place to another, its time period gets increased. How

Let's use Kepler's laws for the inner planets. Use the following distances from the sun to calculate the orbital period for each of these planets. Express your answer in terms of Earth years to two significant figures. Answer for the highlighted planet in each question.

Note: Use Kepler's law directly. Don't just Google the answers, as they will be a little bit different.

When you have calculated them, only submit the value for Earth.

Planet Distance from the sun Period of orbit around the sun

a. Earth 150 million km ___ Earth years
b. Mercury 58 million km ___ Earth years
c. Venus 108 million km ___ Earth years
d. Mars 228 million km ___ Earth years

1. Consider the mechanical energy of a body at rest on the ground at the Earth's equator, at
re = 6400 km.

Consider the mechanical energy of the same body at rest on the ground at the South pole, at re = 6400 km. For this problem, we consider the Earth to be spherical.

(Remember, the object at the equator traces a circular path, the object at the Pole does not.)

G = 6.67×10^−11 Nm^2 kg^−2, and the mass of the Earth is M = 5.97×10^24 kg

a. How much more mechanical energy per kilogram does an object on the ground at the Equator than on the ground at the Pole?
b. E = ___ MJ.kg^−1. (2 sig figs, do not use scientific notation)

1. Consider the mechanical energy of a body in geostationary orbit above the Earth's equator, at rGS=42000 km.

Consider the mechanical energy of the same body on Earth at the South pole, at re = 6400 km. For this problem, we consider the Earth to be spherical. (Remember, the object at the equator is in orbit, the object at the Pole is not in orbit.)

G = 6.67×10^−11 Nm^2 kg^−2, and the mass of the Earth is M = 5.97×10^24 kg

What is the difference in the mechanical energy per kilogram between the two?

E = ___ MJ.kg^−1(to two significant figures, don't use scientific notation)

1. In most calculations, we use W ≈ mg. But we know that, for large changes in altitude, we need to use W∝1/r^2. How far above the Earth's surface can we use W = mg before our systematic error reaches 1%? Use only the information given in this question, and the radius of the Earth re=6400 km. Do not explicitly use G or the mass of the Earth, and do the calculation for the pole so that we don't worry about the effect of centripetal acceleration.

Hint: Does g become larger or smaller with altitude?

Altitude = ___ km (to one significant figure)

1. In the queue at the bus stop, two students (with masses 80 and 60 kg respectively) are standing a distance 0.7 m apart. What is the magnitude of the gravitational force between them?

Treat both objects as point masses, use G = 6.67×10^−11 Nm^2 kg^−2.

F = ___ micro newtons (to one significant figure, and note the units)

2. Ceres is the largest asteroid. Its mean radius is 476 km and its mass is 9.4×10^20 kg. Using this, and G = 6.67×10^−11 Nm^2 kg^−2, and approximating it as a sphere, compute the speed required to escape the gravity on the surface of Ceres.

Speed = ___ m.s^ (to two significant figures, don't use scientific notation)

1. A machine gun fires bullets, each with mass m = 55 g at a speed of u = 450 m.s^−1. The gun fires 5.0 bullets per second. Let's use it as a rocket engine. What thrust does it produce?

Thrust _____ N.

2. A mass breaks suddenly into two parts, masses M and m, with speeds V and v respectively and carrying a total kinetic energy K. If M / m = R = 4.2, what fractions of K does each of the masses carry? (Think about significant figures.)

a) M carries _____ K.

b) m carries _____ K.

Please separate your answers with a comma. For example: if you think that each carries half of the total kinetic energy, write 0.50, 0.50.

1. Let's combine momentum and collisions with some of the other things we've learned. A small block of mass M = 0.201 hangs on the end of a light, in extensible string, length R = 25 cm. A small dart (of mass m = 0.10 kg and small enough to be considered a particle) traveling in the horizontal direction collides with and remains fixed in the block. What is the minimum speed v of the dart such that the combined object completes a circular path around the support point of the block? (Hint: Consider the different parts of this question and analyze each separately before combining them. How fast must the block be traveling at the top? And don't forget significant figures.)

Minimum speed = _____