A planet orbits a star, in a year of length 3.48 x 107 s, in a nearly circular orbit of radius 1.51 x 10¹1 m. With respect to the star, determine (a) the angular speed of the planet, (b) the tangential speed of the planet, and (c) the magnitude of the planet's centripetal acceleration. (a) Number i Units (b) Number i Units (c) Number i Units
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![**Planetary Orbital Dynamics Problem**
A planet orbits a star in a nearly circular orbit with a radius of \( 1.51 \times 10^{11} \) meters, completing one full revolution in a year of length \( 3.48 \times 10^7 \) seconds. With respect to the star, determine the following:
(a) The angular speed of the planet,
(b) The tangential speed of the planet, and
(c) The magnitude of the planet's centripetal acceleration.
### Solution Input Fields:
(a) **Angular Speed**
- **Value:** [ ]
- **Units:** [ ]
(b) **Tangential Speed**
- **Value:** [ ]
- **Units:** [ ]
(c) **Centripetal Acceleration**
- **Value:** [ ]
- **Units:** [ ]
### Instructions:
1. Calculate the angular speed (ω) of the planet using the formula:
\[
ω = \frac{2π}{T}
\]
where \( T \) is the orbital period.
2. Determine the tangential speed (v) of the planet using the formula:
\[
v = ωr
\]
where \( r \) is the radius of the orbit.
3. Calculate the centripetal acceleration (a_c) using the formula:
\[
a_c = \frac{v^2}{r}
\]
### Note:
- Input the calculated value into the respective fields.
- Select the appropriate units from the dropdown options provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd00a953f-93ed-4d4f-930e-90e00b049c18%2F4102cb40-8d56-4eaa-8d21-f8c56763c487%2Ft2rmztb_processed.png&w=3840&q=75)
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