### Continuity and Force Analysis**Problem 16:**Suppose a force exerted by an object on another mass at a distance \( r \) from the center of a planet is given by the function:\[ F(r) = \begin{cases} \frac{2.8r}{R^6}, & \text{if } r < R \\\frac{2.8R}{r^4}, & \text{if } r \ge R \end{cases} \]**Task:**Use the definition of continuity at a point to determine if \( F \) is continuous at \( r = R \).### Explanation:1. **Definition of Continuity:** A function \( f(x) \) is continuous at \( x = c \) if the following three conditions are satisfied: - The function \( f \) is defined at \( x = c \); \( f(c) \) exists. - The limit of \( f(x) \) as \( x \) approaches \( c \) exists. - The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).2. **Checking the Conditions for Continuity at \( r = R \):** - **Existence of \( F(R) \):** \[ F(R) = \frac{2.8R}{R^4} = \frac{2.8R}{R^4} = \frac{2.8}{R^3} \] Thus, \( F(R) \) is defined and equals \( \frac{2.8}{R^3} \). - **Existence of the Limit of \( F(r) \) as \( r \) Approaches \( R \):** - From the left (\( r \to R^- \)): \[ \lim_{r \to R^-} F(r) = \lim_{r \to R^-} \frac{2.8r}{R^6} = \frac{2.8R}{R^6} = \frac{2.8}{R^5} \] - From the right (\( r \to R^+ \)): \[ \lim_{r \to R^+} F(r) = \lim_{r \to R

For any function to continuous it’s LHL, RHL, and function at that point must be equal. The LHL of…

16. Suppose a force exerted by an object on another mass at a distance r from the center of the planet is 2.8r ', if r < R R6 2.8R , if r> R F(r) = r4 at a point to determine if F is continuous at r=R. Use the definition of continuity

16. Suppose a force exerted by an object on another mass at a distance r from the center of the planet is 2.8r ', if r < R R6 2.8R , if r> R F(r) = r4 at a point to determine if F is continuous at r=R. Use the definition of continuity

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

During your winter snowboard run at park city you decide to try a back flip during the movement. When you very first take off on the jump you are moving at 5.4 rad/sec. As you get into the air you tuck to try and complete the trick, this is to ensure that you can decrease your radius of gyration from .51 to .27 during the flip in the air. To make the jump cooler you have a gopro on your helmet that is a distance of 0.25 from your COG where you are rotating. What is the linear velocity of that GoPro during the rotation of your flip. ?
Around 2.5 centuries ago, several physicists of the time came up with the notion of a dark star. This was a star so dense, with so much gravity, that not even light could escape. The calculations used Newtonian mechanics. In class, we calculated the escape speed from the surface of the earth or the distance from the sun, and the mass of the planet or star. Here, the process is partially reversed. Calculate the dark star radius from the mass of the star and the escape speed. Answer in kilometers. c = 3*108 m/s M = 3.2*1030 kg G = 2/3 * 10-10 N*m2/kg2
A satellite in geostationary orbit (also called synchronous orbit) appears to remain stationary in the sky as seen from any particular location on the planet. a.) In the future, there will be need for satellites in synchronous orbit around Mars to aid colonies. At what altitude would such a satellite need to be above the surface of Mars?Assume that the mass of Mars is 6.39 × 10^23 kg, the length of the Martian solar day (i.e., sol) is 24h 39m 35s, the length of the sidereal day is 24h 37m 22s, and the equatorial radius is 3396 km. (Hint: if you haven’t had a physics class before, you can find this by using the fact that the acceleration of an object in circular motion either as v2/r, where v and r are the velocity and radius of the orbit, or as 4Pi 2r/T2 , where T is the period. Use this second equation and Mathematical Insight 4.5 on p. 131 to find r for T=1 day. Make sure to use values for Mars nstead of Earth, as necessary. Alternatively, you can calculate the answer using Newton’s…
Determine the ratio of Earth's gravitational force exerted on an 80-kg person when at Earth's surface and when 1100 km above Earth's surface. The radius of Earth is 6370 km. FE on P,surface FE on Ph 195] ΑΣΦ ?
2. Integrate the equations of motion for a spherical pendulum, i.e., a particle of mass m moving on the surface of a sphere of radius / in a gravitational field, with Lagrangian L = ml² -(0² + psin² 0) + mglcos 0. 2 Hint: Merely set up the integrals, do not evaluate them. You should obtain two integrals w.r.t. de: one determining t and another determining p. Express the first integral in terms of E,Ueff, m, and l. Express the second integral in terms of E,U eff, m, l, and M₂.
An earth-like planet with a mass of 8.00×1024 kg has a space station of mass 4.70×104 kg orbiting it at a distance of 3.00×105 km. What is the gravitational potential energy between the space station and the planet? (We can simplify the Gravitational Constant G to 6.7x10-11 Nm2/kg) J
16-9. Show that the escape velocity from a planet's surface is given by v = (2gR)'/2 where g is the gravitational constant and R the radius of the planet. Given that the gravita- tional constant of the earth is 980 cm/sec² and the radius is 6.4 × 10 cm, calculate the fraction of hydrogen, helium, nitrogen, and oxygen molecules having velocities exceeding the escape velocity.
An object on the Earth has a center of gravity that is 6.0 cm above its base. If the same object was taken to the Moon, where the acceleration due to gravity is 1/6th the value on Earth, how far above the base (cm) would the center of gravity be? O 6.0 O 15 O 3.0 O 2.4 O 36 O 8.4 O 3.6 O 1.0
Problem 9: A thin cylindrical ring starts from rest at a height hj = 75 m. The ring has a radius R = 46 cm and a mass M =2 kg. R h, h,
1. What is the gravitational potential energy in a system of three objects with masses m₁ = 2.0 × 1020 kg, m2 = 4.0 × 1020 kg, and m3 1.0 × 1021 located at r₁ = 2.0 × 10¹3 m 1.0 × 106 m -1.3 × 10¹2 m 1 respectively? 9 r2 = -6.0 × 10¹3 m 2.2 × 10¹² m –5.7 × 10³ m {] 9 T3 = = 1.5 × 10¹1 m −6.5 × 10² m 9.2 × 10¹0 m
A thin spherical shell has radius rA = 4.00 m and mass mA = 20.0 kg. It is concentric with a second thin spherical shell that has radius rB = 6.00 m and mass mB = 40.0 kg. What is the net gravitational force that the two shells exert on a point mass of 0.0200 kg that is a distance r from the common center of the two shells, for (a) r = 2.00 m (inside both shells), (b) r = 5.00 m (in the space between the two shells), and (c) r = 8.00 m (outside both shells)?
Determine the gravitational force F which the titanium sphere exerts on the copper sphere. The value of R is 40 mm. Assume a = 5.0, b = 2.7, 0-38. y 1 j) (108) N aR Titanium Answer: F = (i R Copper bR -x i+ i