Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

문제 69

13장, 문제 69

While visiting Planet Physics, you toss a rock straight up at 11 m/s and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?

검증된 단계별 안내

Step 1: Analyze the motion of the rock. The rock is tossed straight up with an initial velocity of 11 m/s and caught after 2.5 s. Use the kinematic equation for vertical motion under constant acceleration:

y = v

₀t+12at². Here,

y

is the displacement (which is zero since the rock returns to the starting point),

v

₀ is the initial velocity,

t

is the time, and

a

is the acceleration due to gravity on Planet Physics. Solve for

a

Step 2: Use the orbital motion of the cruise ship to determine the mass of Planet Physics. The cruise ship orbits at an altitude equal to the planet's radius, and the orbital period is 230 minutes. Convert the period to seconds and use the formula for orbital velocity:

v = \frac{2}{π} \frac{r}{T}

, where

r

is the radius of the planet and

T

is the orbital period. Then, use the centripetal force equation

F = mv

²r and equate it to the gravitational force

F = \frac{GMm}{r}

² to solve for the mass of the planet.

Step 3: Relate the acceleration due to gravity on the surface of Planet Physics to the planet's mass and radius using the formula

g = \frac{GM}{r}

². Substitute the value of

g

(calculated in Step 1) and solve for the radius

r

Step 4: Combine the results from Step 2 and Step 3 to verify consistency. Ensure that the calculated mass and radius satisfy both the orbital motion and surface gravity conditions.

Step 5: Summarize the process and highlight the relationships between the rock's motion, the cruise ship's orbit, and the planet's physical properties. This reinforces the connection between kinematics, orbital mechanics, and gravitational principles.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this scenario, the initial velocity of the rock (11 m/s) and the time of flight (2.5 s) can be used to analyze its motion using equations of motion, particularly those that relate displacement, velocity, and acceleration.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. The mass of Planet Physics can be determined by analyzing the motion of the rock and applying the gravitational acceleration formula, which relates the gravitational force to the mass of the planet and the distance from its center.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. The cruise ship's orbital period (230 min) provides information about the radius of the planet, as the relationship between the orbital radius and period can be derived from Kepler's laws and Newton's law of gravitation, allowing for the calculation of the planet's radius.

관련 실천

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Two Jupiter-size planets are released from rest 1.0 x 10¹¹ m apart. What are their speeds as they crash together?

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교과서 질문

FIGURE CP13.71 shows a particle of mass m at distance 𝓍 from the center of a very thin cylinder of mass M and length L. The particle is outside the cylinder, so 𝓍 > L/2 . Calculate the gravitational potential energy of these two masses.

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교과서 질문

A 55,000 kg space capsule is in a 28,000-km-diameter circular orbit around the moon. A brief but intense firing of its engine in the forward direction suddenly decreases its speed by 50%. This causes the space capsule to go into an elliptical orbit. What are the space capsule’s (a) maximum and (b) minimum distances from the center of the moon in its new orbit? Hint: You will need to use two conservation laws.

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교과서 질문

A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + Δr, where Δr ≪ r, has the very slightly different period T + ΔT. Show that ΔT/T = (3/2) (Δr/r)

1289

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교과서 질문

Let’s look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii r₁ and r₂, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v'₁ on the inner circular orbit and the velocity v'₁ at the low point on the elliptical orbit that spans the two circular orbits.

1261

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교과서 질문

Let's look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii r₁ and r₂, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?

439

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