Cloverleaf exits are approximately circular. A 1220 kg automobile is traveling 56 mph while taking a cloverleaf exit that has a radius of 35.9 m. Calculate the centripetal acceleration of the car and the centripetal force on the car. Remember that 1 mph = 0.447 m/s.
2.A satellite is in orbit about Earth. Its orbital radius is 7.29×107 m. The mass of the satellite is 4362 kg and the mass of Earth is 5.974×1024 kg. Determine the orbital speed of the satellite in mi/s. 1 mi/s = 1609 m/s.
56 miles/ hr * .447 m/s / mile/hr = 25 m/s
Ac = v^2/R = 625 / 35.9 = 17.4 /s^2 (wow, g is only about 9.81)
F = m Ac
again F = m Ac
this time F = G Mearth m/R^2
and m Ac = m v^2/R
so
G Mearth/R^2 = v^2/R
so
v^2 = G Mearth /R
6.67*10^-11 is G
by the way, the mass of a satellite in orbit does not matter. If you set your coffee cup outside the space station it stays alongside in theory. Notice that m cancels.
To calculate the centripetal acceleration of the car, we can use the formula:
a = v^2 / r
where "a" is the centripetal acceleration, "v" is the velocity, and "r" is the radius.
First, we need to convert the velocity from mph to m/s. Since 1 mph = 0.447 m/s:
v = 56 mph * 0.447 m/s = 25.032 m/s
Substituting the values into the formula:
a = (25.032 m/s)^2 / 35.9 m
a = 627.648 m^2/s^2 / 35.9 m
a ≈ 17.50 m/s^2
So, the centripetal acceleration of the car is approximately 17.50 m/s^2.
To calculate the centripetal force on the car, we can use the formula:
F = m * a
where "F" is the centripetal force, "m" is the mass, and "a" is the centripetal acceleration.
Substituting the values into the formula:
F = 1220 kg * 17.50 m/s^2
F ≈ 21,350 N
Therefore, the centripetal force on the car is approximately 21,350 N.
Now, let's move on to the second question.
To determine the orbital speed of the satellite in mi/s, we can use the formula:
v = √(G * (M / r))
where "v" is the orbital speed, "G" is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2), "M" is the mass of Earth, and "r" is the orbital radius.
First, let's convert the orbital radius from meters to miles:
7.29 × 10^7 m / 1609 m/mi ≈ 45,265 mi
Substituting the values into the formula:
v = √(6.67430 × 10^-11 N m^2/kg^2 * (5.974 × 10^24 kg) / (45,265 mi))
v ≈ √(3.98510 × 10^14 m^3/kg/s^2 * (5.974 × 10^24 kg) / (45,265 mi))
v ≈ √(2.38353 x 10^51 m^3/s^2) ≈ 1.5455 × 10^26 m/s
Now, let's convert the orbital speed from m/s to mi/s:
1.5455 × 10^26 m/s / 1609 m/s/mi ≈ 9.607 × 10^22 mi/s
Therefore, the orbital speed of the satellite is approximately 9.607 × 10^22 mi/s.