**First Examination
**

Physics 104

21 September 1982
#1. A uniform disk (mass ** 0.60 kg**, radius ** 0.15 m**) is mounted at the center
of a (massless) axle which is** 0.30 m ** long, to make a top as shown in the
sketch. A pair of equal and opposite forces, F = **0.2 N**, act at the
ends of the axle (along the **+x** and **- x** directions) while the top
is spinning at **90 rev/s**, as shown. the axle of the top lies along
the **y-axis**. Referring to the coordinate system shown in the sketch,
the **angular speed of precession **has the **direction** given by

**Answer:**

#2. In problem** #1 **above, the **angular speed of precession** of
the top (**in rad/s**) is most nearly equal to

**Answer:**

#3. A simple pendulum takes exactly **0.800 seconds** to complete one
oscillation. Its frequency is most nearly equal to

**Answer:**

#4. The differential equation of motion for a particular simple
harmonic oscillator is

** d**^{2}x / dt^{2} + [ 0.5625 / sec^{2
}] x = 0 ,

where** x **is measured in **cm** and** t** in **seconds**.
The period (in **seconds**) of the oscillator is most nearly equal to

**Answer:**

#5. A mass **M** undergoes **SHM** according to the equation**x
= A cos [ ****w**
** t + ****j**** ]**

The total mechanical energy of the system is equal to

**Answer:**

#6. A mass **M **is on a frictionless horizontal surface, connected by **three**
springs having restoring force constants as shown in the sketch. When set
into **SHM,** the period of the system is.

**Answer:**

#7. The **Lissajous Figure** shown in the sketch is the result of
combining the two **SHM**'s

**x = A**_{x}
cos (w_{x} t )
and y =
A_{y} cos ( w_{y} t + d )

From the figure, the ratio **w**_{x}_{/}**w**_{y}
is equal to

**Answer:**

#8. A solid sphere is suspended at the end of a vertical wire attached
to the ceiling. The sphere is twisted through an angle of **p/2**
**radian** and then released so that it undergoes torsional **SHM**.
If its period is **2p seconds**, then its
maximum angular speed (**in rad/s**) is most nearly equal to

**Answer:**

#9. It is true that, in the strictest sense,

**Answer:**

#10. The mean distance of **Venus** from the sun is **1.08 ´
10**^{9} m, while that of **earth** is **1.50 ´
10**^{9} m.
The number of **earth years** required for **Venus** to complete **one
revolution** around the sun is most nearly equal to

**Answer:**

#11. A simple pendulum on earth's surface has a period of **1.80 sec**.
The same pendulum is found to have a period of **2.00 sec** when operated on
a strange planet. The **gravitational field** at the strange planet's
surface has a magnitude most nearly equal to

**Answer**

#12. A projectile (mass** m**) leaves earth's surface with an initial speed of **v
= Ö(gR)**, straight up, where ** R ** is earth's
radius and ** g ** is the gravitational field at earth's surface, as usual. the
**maximum distance** that the projectile reaches from earth's center is equal to

**Answer:**

#13. A planet has mass **M** distributed uniformly throughout its
volume, a sphere of radius** R**. A small tunnel is bored through the
planet along a diameter, through its center. Assume the planet does not
spin. A ball of mass **m** is released from rest at one end of the
tunnel so that it falls toward the planet's center. At the instant it is
at radius **r** from the center, the **force** acting on the ball is equal
to

**Answer:**

#14. A spaceship (mass **m**) is on the surface of a planet (mass **M**,
radius **R**) which moves in a circular orbit (radius **R**_{s})
about a star (mass** M**_{s}). The planet does not spin.
In order for the spaceship to escape the gravitational fields of both the planet
and the star, it must use an amount of energy equal to

__Note:__ Assume **m << M << M**_{s,} and
take into account of the kinetic energy of the spaceship as it moves with the
planet around the star.

**Answer:**

#15. A mass of **0.20 kg** is connected to the bottom end of a vertically
held, unstretched light spring with a force constant of **5.0 N/m**.
The mass is then released from rest. Take the **X-direction** to
be positive vertically up, and the initial position of the mass to be at **X =
0**. then the position (in m) of the mass exactly **2.500 seconds**
after its release from rest is most nearly equal to

**Answer:**