Problem 44 - University Physics with Modern Physics

Question

There is strong evidence that Europa, a satellite of Jupiter, has a liquid ocean beneath its icy surface. Many scientists think we should land a vehicle there to search for life. Before launching it, we would want to test such a lander under the gravity conditions at the surface of Europa. One way to do this is to put the lander at the end of a rotating arm in an orbiting earth satellite. If the arm is 4.25 m long and pivots about one end, at what angular speed (in rpm) should it spin so that the acceleration of the lander is the same as the acceleration due to gravity at the surface of Europa? The mass of Europa is 4.80 \(\times\) 10\(^{22}\) kg and its diameter is 3120 km.

Step-by-Step Solution

Answer

The required angular speed is approximately 5.31 rpm.

Step 1: Calculate Europa's Surface Gravity

The formula for gravitational acceleration is given by \[ g = \frac{G \cdot M}{R^2} \]where:- \( G = 6.674 \times 10^{-11} \text{ m}^3 / (\text{kg} \cdot \text{s}^2) \) (the universal gravitational constant),- \( M = 4.80 \times 10^{22} \text{ kg} \) (mass of Europa), and - \( R \) is the radius of Europa, which is half of its diameter: \( 1560 \times 10^3 \text{ m} \).Plug in these values to compute \( g \): \[ g_E = \frac{6.674 \times 10^{-11} \times 4.80 \times 10^{22}}{(1560 \times 10^3)^2} \approx 1.314 \text{ m/s}^2 \].

Step 2: Relate Centripetal Acceleration to Angular Speed

To achieve the same acceleration using a rotating arm, the centripetal acceleration formula is used:\[ a_c = r \cdot \omega^2 \]where:- \( a_c = g_E = 1.314 \text{ m/s}^2 \) (given from Step 1)- \( r = 4.25 \text{ m} \) (length of the arm)- \( \omega \) is the angular speed in radians per second.We need to find \( \omega \). Rearrange:\[ \omega^2 = \frac{a_c}{r} \text{, so}\]\[ \omega = \sqrt{\frac{1.314}{4.25}} \].

Step 3: Calculate Angular Speed in Radians per Second

Continue from Step 2:\[ \omega = \sqrt{\frac{1.314}{4.25}} \approx 0.556 \text{ rad/s} \].

Step 4: Convert Angular Speed to Revolutions per Minute (rpm)

To convert from radians per second to revolutions per minute (rpm), use the conversion factors:1. 1 revolution = \( 2\pi \) radians,2. 1 minute = 60 seconds.Thus:\[ \text{rpm} = \omega \times \frac{1 \text{ rev}}{2 \pi \text{ rad}} \times \frac{60 \text{ s}}{1 \text{ min}} \approx 0.556 \times \frac{60}{2\pi} \approx 5.31 \text{ rpm} \].

Key Concepts

Centripetal Acceleration

Centripetal acceleration is an essential concept in physics whenever an object is in circular motion, like a car turning around a bend or a lander being swung by an arm. This type of acceleration always points towards the center of the circle along which the object is moving. It is what keeps the object in its circular path instead of flying off in a straight line. To calculate centripetal acceleration, we use the formula:\[ a_c = r \cdot \omega^2 \]where:- \( a_c \) is the centripetal acceleration,- \( r \) is the radius of the circular path, and - \( \omega \) is the angular speed in radians per second.The centripetal acceleration depends on how fast the object is spinning (speed) and how far away it is from the center (radius). The faster the spin or the longer the arm, the greater the acceleration experienced.

Universal Gravitational Constant

The Universal Gravitational Constant, denoted as \( G \), is a key component in the law of universal gravitation, developed by Sir Isaac Newton. This constant characterizes the strength of gravitational attraction between two bodies. Its value is always the same: \[ G = 6.674 \times 10^{-11} \text{ m}^3 / (\text{kg} \cdot \text{s}^2) \]It allows us to calculate the gravitational force between two masses. In the context of finding surface gravity like on Europa, \( G \) helps to determine the gravitational pull by: \[ F = \frac{G \cdot M_1 \cdot M_2}{r^2} \]where \( M_1 \) and \( M_2 \) are the masses of the two objects and \( r \) is the distance between their centers. This formula shows that gravitational force decreases with the square of the distance, making \( G \) an essential tool for understanding gravitational interactions in the universe.

Europa's Surface Gravity

Europa's surface gravity is the acceleration due to gravity experienced by objects on its surface. It is crucial to understand when planning missions, as this gravity determines how objects like spacecraft will behave when they land. To find this, we employ the formula for gravitational acceleration:\[ g = \frac{G \cdot M}{R^2} \]where:- \( G \) is the universal gravitational constant,- \( M \) is the mass of Europa, and- \( R \) is the radius of Europa.Plugging in the values as provided, Europa's surface gravity \( g_E \) is calculated to be about \( 1.314 \text{ m/s}^2 \). This is significantly weaker than Earth's gravity, which is roughly \( 9.81 \text{ m/s}^2 \). Knowing Europa's surface gravity helps engineers design spacecraft that can maneuver and land safely on the icy moon, even under these reduced gravitational forces.