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Chapter 15: Q. 47 (page 417)
A block hangs in equilibrium from a vertical spring. When a second identical block is added, the original block sags by . What is the oscillation frequency of the two-block system?
The two block system have oscillation frequency of.
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Suppose a large spherical object, such as a planet, with radius and mass has a narrow tunnel passing diametrically through it. A particle of mass is inside the tunnel at a distance from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius ; there is no net gravitational force from the mass in the spherical shell with .
Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of , and any necessary constants.
You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with . Suppose an intrepid astronaut exploring a diameter, asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?
A block oscillating on a spring has period . What is the period if:
a. The block's mass is doubledExplain. Note that you do not know the value of either or , so do not assume any particular values for them. The required analysis involves thinking about ratios.
b. The value of the spring constant is quadrupled?
c. The oscillation amplitude is doubled while and are unchanged?
shows a uniform rod pivoted at one end. The other end is attached to a horizontal spring. The spring is neither stretched nor compressed when the rod hangs straight down. What is the rod鈥檚 oscillation period? You can assume that the rod鈥檚 angle from vertical is always small.
The two blocks in FIGURE P15.52 oscillate on a frictionless surface with a period of . The upper block just begins to slip when the amplitude is increased to . What is the coefficient of static friction between the two blocks?
FIGURE P15.52
It is said that Galileo discovered a basic principle of the pendulum鈥�
that the period is independent of the amplitude鈥攂y using
his pulse to time the period of swinging lamps in the cathedral
as they swayed in the breeze. Suppose that one oscillation of a
swinging lamp takes.
a. How long is the lamp chain?
b. What maximum speed does the lamp have if its maximum
angle from vertical is ?
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