Explanation:
(a) Draw a free body diagram of the cylinder at the top of the loop. At the minimum speed, the normal force is 0, so the only force is weight pulling down.
Sum of forces in the centripetal direction:
∑F = ma
mg = mv²/RL
v = √(g RL)
(b) Energy is conserved.
EE = KE + RE + PE
½ kd² = ½ mv² + ½ Iω² + mgh
kd² = mv² + Iω² + 2mgh
kd² = mv² + (m RC²) ω² + 2mg (2 RL)
kd² = mv² + m RC²ω² + 4mg RL
kd² = mv² + mv² + 4mg RL
kd² = 2mv² + 4mg RL
kd² = 2m (v² + 2g RL)
d² = 2m (v² + 2g RL) / k
d = √[2m (v² + 2g RL) / k]
B. Elastic potential to kinetic energy
The elastic potential energy in the slingshot will be transferred to the stone as kinetic energy as the stone is launched.
A.
if you have seen a newton's cradle this will make sense.
in order for both of them to travel at the same speed, the balls need to have the same mass and the speed to begin with tocontinue to travel at the same speed because mass can affect the impact of the force on the balls by each other, causing each ball to have different speeds.
The total flux through the cylinder is zero.
In fact, the electric flux through a surface (for a uniform electric field) is given by:
where
E is the intensity of the electric field
A is the surface
is the angle between the direction of E and the perpendicular to the surface, whose direction is always outwards of the surface.
We can ignore the lateral surface of the cylinder, since the electric field is parallel to it, therefore the flux through the lateral surface of the cylinder is zero (because 
and 
).
On the other two surfaces, the flux is equal and with opposite sign. In fact, on the first surface the flux will be
where r is the radius, and where we have taken 
since the perpendicular to the surface is parallel to the direction of the electric field, so 
. On the second surface, however, the perpendicular to the surface is opposite to the electric field, so 
and 
, therefore the flux is
And the net flux through the cylinder is
The ability for CO2 to warm the surface of a planet through the absorption of infrared radiation is well known. What is much less appreciated, however, is just how effective of a gas it is in maintaining the greenhouse framework that helps to characterize the modern climate.
