Physics 1

An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has a mass of M1= 1.90 x 1027kg and radius R1= 7.14 x 107m. A. If the radius of the planned orbit is R, use Newton’s laws to show each of the following.

The orbital speed of the planned satellite is give by v= $\sqrt{\frac{GM_1}{R}}$

The period of the orbit is given by T= $\sqrt{\frac{4\pi^2R^3}{GM_1}}$

B. The explorer wants the satellite’s orbit to be synchronized with Jupiter’s rotation. This requires an equatorial orbit whose period equals Jupiter’s rotation period of 9 hr 51 min = 3.55 x 105s. Determine the required orbital radius in meters. C. Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs form the desired value in each of the following ways, sketch the resulting orbit on the figure. ( J is the center of Jupiter, the dashed circle is the desired orbit, and Pis the injection point.) Also, describe the resulting orbit qualitatively but specifically

When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius.

When the satellite is at the desired altitude over the equation, its velocity vector has the correct direction but the speed is slightly slower than the correct speed for a circular orbit of that radius.

A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown below. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible.

D. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part (C), Greater, Equal, or Less?

A uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows.

C. Determine the linear speed of the mass at the end of the rod for the instant the rod is in the horizontal position.

A solid disk of unknown mass and known radius R is used as a pulley in a lab experiment as shown below. A small block of mass m is attached to a string, the other end of which is attached to the pulley and wrapped around it several times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor.

D. The value of acceleration found in (B)iii, along with numerical values for the given quantities and your answer to (C), can be used to determine the rotational inertia of the pulley. The pulley is removed from its support and its rotational inertia is found to be greater than this value. Give one explanation for this discrepancy.

A uniform rod of mass M and length L is attached to a pivot of neglible friction as shown below. The pivot is located at a distance $\frac{L}{3}$ from the left end of the rod. Express all answers in terms of the given quantities and fundamental constants.

A system consists of a ball of mass $M_2$ and a uniform rod of mass $M_1$ and length $d$. The rod is attached to a horizontal frictionless table by a pivot at point $P$ and initially rotates at an angular speed $\omega$, pictured below on the left. The rotational inertia of the rod about point $P$ is $\frac{1}{3}M_1d^2$ . The rod strikes the ball, which is initially at rest. As a result of this collision, the rod is stopped and the ball moves in the direction shown on the right. Express all answers in terms of $M_1$ , $M_2$ , $\omega$ , $d$ , and fundamental constants.

D. A new ball with the same mass $M_1$ as the rod is now placed a distance $x$ from the pivot, as shown above. Again assuming the collision is elastic, for what value $x$ will the rod stop moving after hitting the ball?

In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth. Assume a circular orbit with a period of 1.18 x 10 $^2$ minutes = 7.08 x 10 $^3$ m/s. The mass of the GS is 930 kg and the radius of Mars is 3.43 x 10$^6$ m .

A. Calculate the radius of the GS orbit.

B. Calculate the mass of Mars.

C. Calculate the total mechanical energy of the GS in this orbit.

D. If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period of the GS be greater than or less than the given period?

E. In fact, the orbit of the GS entered was slightly eliptical with its closest approach to Mars at 3.71 x 10 $^5$ m above the surface and its furthest distance at 4.36 x 10 $^5$ m above the surface. If the speed of the GS at closest approach is 3.40 x 10 $^3$ m/s, calculate the speed at the furthest point of the orbit.

An oscillator consists of a block of mass 0.500 kg connected to a spring. When set into oscillation with amplitude 35.0 cm, the oscillator repeates its motion every 0.500 s. Find the:

A. period

B. frequency

C. angular frequency

D. spring constant

E. maximum speed

F. magnitude of maximum force on the block from the spring

An oscillator consists of a block attached to a spring( k = 400 N/m). At some time t, the position (measured from the system's equilibrium location), velocity, and acceleration of the block are x = 0.100m, v = -13.6 m/s, and a = -123 m/s$^2$ . Calculate the

A. frequency of oscillation

B. mass of the block

C. amplitude of the motion

A block is on a horizontal surface ( a shake table ) that is moving back and forth horizontally with simple harmonic motion of frequency 2.0 Hz. The coefficient of static friction between block and surface is 0.50. How great can the amplitude of the simple harmonic motion be if the block is not to slip along the surface?

Two blocks (m = 1.8 kg and M = 10 kg) and a spring ( k = 200 N/m ) are arranged on a horizontal frictionless surface. The coefficient of static friction between the two blocks is 0.40. What amplitude of simple harmonic motion of the spring-block system puts the smaller block on the verge of slipping over the larger block?

Find the mechanical energy of a block-spring system having a spring constant of 1.3 N/cm and an oscillation amplitude of 2.4 cm.

A 10 g particle undergoes simple harmonic motion with an amplitude of 2.0 mm, a maximum acceleration of 8 x 10$^3$ m/s$^2$ , and an unknown phase constant. What are

A. the period of the motion?

B. the maximum speed of the particle?

C. the total mechanical energy of the oscillator?

D. the magnitude of the force on the particle at its maximum and half its maximum displacement?

A block of mass M, 5.4 kg, at rest on a horizontal frictionless table is attached to a rigid support by a spring constant k, 6000 N/m. A bullet of mass m, 9.5 g, and velocity of magnitude 630 m/s strikes and is embedded in the block. Assuming the compression of the spring is negligible until the bullet is embedded, determine:

B. the amplitude of the resulting simple harmonic motion

An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching it a distance D as shown above. A piece of clay, also of mass M, is then dropped from a height H onto the pan and sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants.

E. The clay is now removed from the pan and the pan is returned to equilibrium at the end of the spring. A rubber ball, also of mass M, is dropped from the same height H onto the pan, and after the collisions is caught in mid air before hitting anything else. Indicate whether the period of the resulting simple harmonic motion of the pan is greater than, less than, or the same as it was in part (C).