A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to
1 answer:
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a) 0.0028 rad/s
b)
c)
Explanation:
a)
When an object is in circular motion, the angular speed of the object is the rate of change of its angular position. In formula, it is given by
where
is the angular displacement
t is the time interval
The angular speed of an object in circular motion can also be written as
(1)
where
v is the linear speed of the object
r is the radius of the orbit
For the spaceship in this problem we have:
is the linear speed, converted into m/s,
is the radius of the orbit
Subsituting into eq(1), we find the angular speed of the spaceship:
b)
When an object is in circular motion, its direction is constantly changing, therefore the object is accelerating; in particular, there is a component of the acceleration acting towards the centre of the orbit: this is called centripetal acceleration, or radial acceleration.
The magnitude of the radial acceleration is given by
where
is the angular speed
is the radius of the orbit
For the spaceship in the problem, we have
is the angular speed
is the radius of the orbit
Substittuing into the equation above, we find the radial acceleration:
c)
When an object is in circular motion, it can also have a component of the acceleration in the direction tangential to its motion: this component is called tangential acceleration.
The tangential acceleration is given by
where
is the change in the linear speed
is the time interval
In this problem, the spaceship is moving with constant linear speed equal to
Therefore, its linear speed is not changing, so the change in linear speed is zero:
And therefore, the tangential acceleration is zero as well: