SEPARATION OF VARIABLES - SPHERICAL The potential on the surface of a sphere (radius R) is given by V=V0 cos(2). (Assume V(r=[i
nfinity])=0. Also assume there is no charge in or outside, it's ALL on the surface!) a) Find the potential everywhere inside and outside this sphere. (Hint: Can you express cos(2) as a simple linear combination of Legendre polynomials? ) b) Find the charge density () on the surface of the sphere in part a. Sketch or draw it!
1 answer:
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Answer:
a) The car will reach a height of 45.9 m.
b) The amount of thermal energy generated is 173382 J.
c) The magnitude of the force of friction is 417.8 N.
Explanation:
Hi there!
a) In this problem, we have to use the conservation of energy. The energy conservation theorem states that the energy of a system remains constant. Energy can´t be created nor destroyed, only transformed. In the case of the car, the initial kinetic energy is transformed into potential energy as the car´s height increases while coasting up the hill.
Then, all the initial kinetic energy (KE) will be transformed into potential energy (PE) (only if there is no friction).
The equation of KE is the following:
KE = 1/2 · m · v²
Where:
m = mass of the car.
v = speed of the car.
The equation of PE is the following:
PE = m · g · h
Where:
m = mass of the car.
g = acceleration due to gravity.
h = height at which the car is located.
Since work done by friction is negligible, we can assume that all the initial kinetic energy will be transformed into potential energy. Then:
KE at the bottom of the hill = PE at the top of the hill
1/2 · m · v² = m · g · h
Solving for h:
1/2 · v² / g = h
Let´s convert the speed unit into m/s:
108 km/h · 1000 m/ 1 km · 1 h / 3600 s = 30 m/s
Now, let´s calculate h:
h = 1/2 · (30 m/s)² / 9.8 m/s²
h = 45.9 m
The car will reach a height of 45.9 m.
b) In this case, all the kinetic energy is not transformed into potential energy because some energy is transformed into thermal energy due to friction. The thermal energy generated is equal to the work done by friction. Then:
KE at the bottom of the hill = PE + work done by friction
KE = PE + Wfr (where Wfr is the work done by friction).
1/2 · m · v² = m · g · h + Wfr
1/2 · m · v² - m · g · h = Wfr
1/2 · 710 kg · (30 m/s)² - 710 kg · 9.8 m/s² · 21 m = Wfr
Wfr = 173382 J
The amount of thermal energy generated is 173382 J.
c) The work done by friction is calculated as follows:
Wfr = Ffr · Δx
Where:
Ffr = friction force.
Δx = traveled distance
Please, see the attached figure to notice that the traveled distance can be calculated by trigonometry using this trigonometric rule of right triangles:
sin angle = opposite side / hypotenuse
In our case:
sin 2.9° = h / Δx
Δx = h / sin 2.9°
Δx = 21 m / sin 2.9° = 415 m
Then, solving for the friction force using the equation of the work done by friction:
Wfr = Ffr · Δx
Wfr / Δx = Ffr
173382 J / 415 m = Ffr
Ffr = 417.8 N
The magnitude of the force of friction is 417.8 N
