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Lemur [1.5K]
3 years ago
12

An electron remains suspended between the surface of the Earth (assumed neutral) and a fixed positive point charge, at a distanc

e of 8.88 m from the point charge. The acceleration of gravity is 9.8 m/s^2 and the value of Coulomb's constant is 8.98755 x 10^9 Nm^2/C^2. Determine the charge required for this to happen. Mass of electron is 9.10939 x 10^-31 kg.
Physics
1 answer:
DaniilM [7]3 years ago
5 0

To solve this problem we will apply the concept related to the balance of forces. Here the electrostatic force defined by Coulomb's laws must be proportional to the weight of the electron, defined by Newton's second law, so that such balance exists, both can be described as:

F_e = \frac{kq_1q_2}{d^2}

Here,

q_1 = Charge required

q_2 =Charge of electron

d = Distance

k = Coulomb's constant

F_w = mg

Here,

m = mass

g = Gravitational acceleration

In equilibrium the sum  of Force is zero, then,

\sum F = 0

F_e = F_w

\frac{kq_1q_2}{d^2} = mg

\frac{(8.98755*10^9 )(q_1)(1.6*10^{-19})}{8.8^2} = (9.8)(9.10939*10^{-31} )

q_1 = 4.8075*10^{-19}C

Therefore the charge required for this to happen is 4.8075*10^{-19}C

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A projectile is launched at an angle of 36.7 degrees above the horizontal with an initial speed of 175 m/s and lands at the same
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Answer:

a) The maximum height reached by the projectile is 558 m.

b) The projectile was 21.3 s in the air.

Explanation:

The position and velocity of the projectile at any time "t" is given by the following vectors:

r = (x0 + v0 · t · cos α, y0 + v0 · t · sin α + 1/2 · g · t²)

v = (v0 · cos α, v0 · sin α + g · t)

Where:

r = position vector at time "t"

x0 = initial horizontal position

v0 = initial velocity

t = time

α = launching angle

y0 = initial vertical position

g = acceleration due to gravity (-9.80 m/s² considering the upward direction as positive).

v = velocity vector at time t

a) Notice in the figure that at maximum height the velocity vector is horizontal. That means that the y-component of the velocity (vy) at that time is 0. Using this, we can find the time at which the projectile is at maximum height:

vy = v0 · sin α + g · t

0 = 175 m/s · sin 36.7° - 9.80 m/s² · t

-  175 m/s · sin 36.7° /  - 9.80 m/s² = t

t = 10.7 s

Now, we have to find the magnitude of the y-component of the vector position at that time to obtain the maximum height (In the figure, the vector position at t = 10.7 s is r1 and its y-component is r1y).

Notice in the figure that the frame of reference is located at the launching point, so that y0 = 0.

y = y0 + v0 · t · sin α + 1/2 · g · t²

y = 175 m/s · 10.7 s · sin 36.7° - 1/2 · 9.8 m/s² · (10.7 s)²

y = 558 m

The maximum height reached by the projectile is 558 m

b) Since the motion of the projectile is parabolic and the acceleration is the same during all the trajectory, the time of flight will be twice the time it takes the projectile to reach the maximum height. Then, the time of flight of the projectile will be (2 · 10.7 s) 21.4 s. However, let´s calculate it using the equation for the position of the projectile.

We know that at final time the y-component of the vector position (r final in the figure) is 0 (because the vector is horizontal, see figure). Then:

y = y0 + v0 · t · sin α + 1/2 · g · t²

0 = 175 m/s · t · sin 36.7° - 1/2 · 9.8 m/s² · t²

0 = t (175 m/s ·  sin 36.7 - 1/2 · 9.8 m/s² · t)

0 = 175 m/s ·  sin 36.7 - 1/2 · 9.8 m/s² · t

-  175 m/s ·  sin 36.7 / -(1/2 · 9.8 m/s²) = t

t = 21.3 s

The projectile was 21.3 s in the air.

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