Question

A spring with constant k1 is attached vertically to the ceiling and a sphere of mass m1 hangs from the other end of the spring. Another spring, this one with constant k2 is attached vertically to the first sphere. Finally a second sphere (mass m2) is attached to the lower end of the second spring. Assuming that the spheres can only move vertically and using y1 and y2 as coordinates measured from the equilibrium position of each sphere, show that the equations of motion can be expressed as M ¨~y = −K~y where ~y = y1 y2 , and M and K are 2 × 2 matrices. Find those matrices as well

Answer #1

A mass hangs vertically from a spring that is attached to the
ceiling, and oscillates. If the kinetic energy of the mass is
decreasing, what must be true about the mass's motion?

A mass hangs vertically from a spring that is attached to the
ceiling, and oscillates. If the kinetic energy of the mass is
decreasing, what must be true about the mass's motion?
The mass must be moving towards the equilibrium position.
The mass must be moving upward.
The mass must be moving away from the equilibrium
position.
The mass must be moving downward.

A spring with an unstrained length of 0.075 m and a spring
constant of 2.2 N/m hangs vertically downward from the ceiling. A
uniform electric field directed upward fills the region containing
the spring. A sphere with a mass of 5.9 × 10-3 kg and a net charge
of +7.2 μC is attached to the lower end of the spring. The spring
is released slowly, until it reaches equilibrium. The equilibrium
length of the spring is 0.063 m. What is...

A spring with spring constant 17 N/m hangs from the ceiling. A
ball is attached to the spring and allowed to come to rest. It is
then pulled down 9.5 cm and released. The ball makes 32
oscillations in 16 s seconds. What is its maximum speed?

A 12 cm long spring is attached to the ceiling. When a 2.4 kg
mass is hung from it, the spring stretches to a length of 17
cm.
(a) What is the spring constant k?
N/m
(b) How long is the spring when a 3.2 kg mass is suspended from
it?
cm

Consider a block attached to one end of an ideal spring with
spring constant k=10 N/m. The other end of the spring is fixed to
the ceiling. The block is moving vertically with simple harmonic
oscillations. During the oscillations, the speed of the block
reaches a maximum value of 10 m/s and the maximum acceleration of
the block is 50 m/s2.
What is the mass of the block? Express your answer in
units of kg, but enter only the numeric...

A spring mass harmonic oscillator consists of a 0.2kg mass
sphere connected vertically with a spring of negligible mass and
force constant of 6kN / m. The spring is released from rest 3cm
from the equilibrium position. Calculate:
(a) The energy of the spring,
(b) The potential energy a when the compression of the spring is
1/3 of the amplitude,
(c) Kinetic energy at this time.

1 A massless spring with spring constant ? hangs from the
ceiling with a small object of mass ? attached to its lower end.
The object is initially held at the spring’s rest position. The
object is then released oscillates up and down, with its lowest
position being 10 ?? below the point from which it was released. a)
What is the value of the ratio of the spring constant of the spring
over the mass attached? That is, what...

A block of unknown mass is attached to a spring with a spring
constant of 7.00 N/m and undergoes simple harmonic motion with an
amplitude of 11.5 cm. When the block is halfway between its
equilibrium position and the end point, its speed is measured to be
25.0 cm/s.
(a) Calculate the mass of the block.
kg
(b) Calculate the period of the motion.
s
(c) Calculate the maximum acceleration of the block.
m/s2

A block of unknown mass is attached to a spring with a spring
constant of 5.50 N/m and undergoes simple harmonic motion with an
amplitude of 10.0 cm. When the block is halfway between its
equilibrium position and the end point, its speed is measured to be
27.0 cm/s.
(a) Calculate the mass of the block.
(b) Calculate the period of the motion.
(c) Calculate the maximum acceleration of the block.

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