IPhO第届国际物理奥林匹克竞赛理论试题与解答
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= +
(A.14)
and the acceleration
2
rel = rel
.
(A.15)
At the lowest point of the puck trajectory the acceleration of the cylinder axis is equal to zero,
thin hollow thin cylinder of mass and of radius . Initially, the cylinder
rests on the horizontal plane and the puck is located at the height above
acceleration is found as
= sin − .
(A.2)
Since the cylinder moves along the plane without sliding its
angular acceleration is obtained as
= /
the plane as shown in the figure on the left. Find the interaction force
between the puck and the cylinder at the moment when the puck passes
the lowest point of its trajectory. Assume that the friction between the
capacitor of capacitance 2 carries the electric charge 0 , a capacitor of
capacitance is uncharged, and there are no electric currents in both coils of
formula for the frequency of the small radial oscillations of the bubble and evaluate it under the
assumption that the heat capacity of the soap film is much greater than the heat capacity of the gas in the
puck and the inner surface of the cylinder is absent, and the cylinder
moves on the plane without slipping. The free fall acceleration is .
Part B (3 points)
instantly shorted. Find the maximum current max through the switch thereafter.
Theoretical competition. Tuesday, 15 July 2014
1/4
Problem 1
Solution
Part A
=
,
(A.12)
(2+ )
Biblioteka Baidu
(2+ )
.
(A.13)
In the reference frame sliding progressively along with the cylinder axis, the puck moves in a circle
of radius and, at the lowest point of its trajectory, have the velocity
the gas pressure inside the bubble is defined by
4
= ,
(B.3)
thus, the equation of any equilibrium process with the gas in the bubble is a polytrope of the form
2
surface, caused by surface tension of the interface between liquid and gas, so that ∆ = .
Part C (3 points)
Initially, a switch is unshorted in the circuit shown in the figure on the right, a
Consider the forces acting on the puck and the cylinder and
depicted in the figure on the right. The puck is subject to the
gravity force and the reaction force from the cylinder . The
therefore, the puck acceleration in the laboratory reference frame is also given by (A.15).
2
− = .
then the interaction force between the puck and the cylinder is finally found as
Theoretical competition. Tuesday, 15 July 2014
1/1
Problem 1 (9 points)
This problem consists of three independent parts.
Part A (3 points)
A small puck of mass is carefully placed onto the inner surface of the
cylinder is subject to the gravity force , the reaction force from
the plane 1 , the friction force and the pressure force from the
puck ′ = −. The idea is to write the horizontal projections of
the equations of motion. It is written for the puck as follows
= sin ,
(A.1)
where is the horizontal projection of the puck acceleration.
For the cylinder the equation of motion with the
inductance and 2, respectively. The capacitor starts to discharge and at the
moment when the current in the coils reaches its maximum value, the switch is
bubble. Assume that the thermal equilibrium inside the bubble is reached much faster than the period of
oscillations.
Hint: Laplace showed that there is pressure difference between inside and outside of a curved
= ,
(A.10)
since it does not slide over the plane.
Solving (A.8)-(A.10) results in velocities at the lowest point of the puck trajectory written as
=2
bubble is found as
= + ,
(B.1)
where the molar heat capacity at arbitrary process is as follows
1
= = + .
(B.2)
Here stands for the molar heat capacity of the gas at constant volume, designates its pressure, is the
total amount of moles of gas in the bubble, and denote the volume and temperature of the gas,
respectively.
Evaluate the derivative standing on the right hand side of (B.2). According to the Laplace formula,
(A.3)
Then the equation of rotational motion around the center of mass of the cylinder takes the form
= ,
(A.4)
where the inertia moment of the hollow cylinder is given by
g
= 1.10 3 . 1) Find formula for the molar heat capacity of the gas in the bubble for such a process when
cm
the gas is heated so slowly that the bubble remains in a mechanical equilibrium and evaluate it; 2) Find
A bubble of radius = 5.00 cm, containing a diatomic ideal gas, has the soap film of thickness ℎ =
N
10.0 μm and is placed in vacuum. The soap film has the surface tension = 4.00 ∙ 10−2 m and the density
= 3 1 + 3 .
(A.16)
(A.17)
Theoretical competition. Tuesday, 15 July 2014
2/4
Part B
1) According to the first law of thermodynamics, the amount of heat transmitted to the gas in the
= 2 .
(A.5)
Solving (A.2)-(A.5) yields
2 = sin .
(A.6)
From equations (A.1) and (A.6) it is easily concluded that
= 2.
(A.7)
Since the initial velocities of the puck and of the cylinder are both equal to zero, then, it follows from
(A.7) after integrating that
= 2.
(A.8)
It is obvious that the conservation law for the system is written as
2
2
2
= 2 + 2 + 2 ,
(A.9)
where the angular velocity of the cylinder is found to be
(A.14)
and the acceleration
2
rel = rel
.
(A.15)
At the lowest point of the puck trajectory the acceleration of the cylinder axis is equal to zero,
thin hollow thin cylinder of mass and of radius . Initially, the cylinder
rests on the horizontal plane and the puck is located at the height above
acceleration is found as
= sin − .
(A.2)
Since the cylinder moves along the plane without sliding its
angular acceleration is obtained as
= /
the plane as shown in the figure on the left. Find the interaction force
between the puck and the cylinder at the moment when the puck passes
the lowest point of its trajectory. Assume that the friction between the
capacitor of capacitance 2 carries the electric charge 0 , a capacitor of
capacitance is uncharged, and there are no electric currents in both coils of
formula for the frequency of the small radial oscillations of the bubble and evaluate it under the
assumption that the heat capacity of the soap film is much greater than the heat capacity of the gas in the
puck and the inner surface of the cylinder is absent, and the cylinder
moves on the plane without slipping. The free fall acceleration is .
Part B (3 points)
instantly shorted. Find the maximum current max through the switch thereafter.
Theoretical competition. Tuesday, 15 July 2014
1/4
Problem 1
Solution
Part A
=
,
(A.12)
(2+ )
Biblioteka Baidu
(2+ )
.
(A.13)
In the reference frame sliding progressively along with the cylinder axis, the puck moves in a circle
of radius and, at the lowest point of its trajectory, have the velocity
the gas pressure inside the bubble is defined by
4
= ,
(B.3)
thus, the equation of any equilibrium process with the gas in the bubble is a polytrope of the form
2
surface, caused by surface tension of the interface between liquid and gas, so that ∆ = .
Part C (3 points)
Initially, a switch is unshorted in the circuit shown in the figure on the right, a
Consider the forces acting on the puck and the cylinder and
depicted in the figure on the right. The puck is subject to the
gravity force and the reaction force from the cylinder . The
therefore, the puck acceleration in the laboratory reference frame is also given by (A.15).
2
− = .
then the interaction force between the puck and the cylinder is finally found as
Theoretical competition. Tuesday, 15 July 2014
1/1
Problem 1 (9 points)
This problem consists of three independent parts.
Part A (3 points)
A small puck of mass is carefully placed onto the inner surface of the
cylinder is subject to the gravity force , the reaction force from
the plane 1 , the friction force and the pressure force from the
puck ′ = −. The idea is to write the horizontal projections of
the equations of motion. It is written for the puck as follows
= sin ,
(A.1)
where is the horizontal projection of the puck acceleration.
For the cylinder the equation of motion with the
inductance and 2, respectively. The capacitor starts to discharge and at the
moment when the current in the coils reaches its maximum value, the switch is
bubble. Assume that the thermal equilibrium inside the bubble is reached much faster than the period of
oscillations.
Hint: Laplace showed that there is pressure difference between inside and outside of a curved
= ,
(A.10)
since it does not slide over the plane.
Solving (A.8)-(A.10) results in velocities at the lowest point of the puck trajectory written as
=2
bubble is found as
= + ,
(B.1)
where the molar heat capacity at arbitrary process is as follows
1
= = + .
(B.2)
Here stands for the molar heat capacity of the gas at constant volume, designates its pressure, is the
total amount of moles of gas in the bubble, and denote the volume and temperature of the gas,
respectively.
Evaluate the derivative standing on the right hand side of (B.2). According to the Laplace formula,
(A.3)
Then the equation of rotational motion around the center of mass of the cylinder takes the form
= ,
(A.4)
where the inertia moment of the hollow cylinder is given by
g
= 1.10 3 . 1) Find formula for the molar heat capacity of the gas in the bubble for such a process when
cm
the gas is heated so slowly that the bubble remains in a mechanical equilibrium and evaluate it; 2) Find
A bubble of radius = 5.00 cm, containing a diatomic ideal gas, has the soap film of thickness ℎ =
N
10.0 μm and is placed in vacuum. The soap film has the surface tension = 4.00 ∙ 10−2 m and the density
= 3 1 + 3 .
(A.16)
(A.17)
Theoretical competition. Tuesday, 15 July 2014
2/4
Part B
1) According to the first law of thermodynamics, the amount of heat transmitted to the gas in the
= 2 .
(A.5)
Solving (A.2)-(A.5) yields
2 = sin .
(A.6)
From equations (A.1) and (A.6) it is easily concluded that
= 2.
(A.7)
Since the initial velocities of the puck and of the cylinder are both equal to zero, then, it follows from
(A.7) after integrating that
= 2.
(A.8)
It is obvious that the conservation law for the system is written as
2
2
2
= 2 + 2 + 2 ,
(A.9)
where the angular velocity of the cylinder is found to be