Question

two artificial satellites, 1 and 2, orbit the earth in circular orbits having radii r₁ and r₂, respec-tively, as shown above. if r₂ = 2r₁, the accelerations a₂ and a₁ of the two satellites are related bywhich of the following?(a) a₂ = 4a₁ (b) a₂ = 2a₁ (c) a₂ = a₁ (d) a₂ = a₁/2 (e) a₂ = a₁/4two identical stars, a fixed distance d apart, revolve in a circle about their mutual center ofmass, as shown above. each star has mass m and speed v. g is the universal gravitationalconstant. which of the following is a correct relationship among these quantities?(a) v² = gm/d (b) v² = gm/2d (c) v² = gm/d² (d) v² = mgd (e) v² = 2gm²/da particle is moving in a circle of radius 2 meters according to the relation θ = 3t² + 2t, where θ ismeasured in radians and t in seconds.the tangential acceleration of the particle at t = 4 seconds is(a) 6 m/s² (b) 12 m/s² (c) 24 m/s² (d) 26 m/s² (e) 52 m/s²the radial acceleration of the particle at t = 4 seconds is(a) 26 m/s² (b) 52 m/s² (c) 1352 m/s² (d) 2704 m/s² (e) 5408 m/s²

Explanation:

Problem 8

Step1: Find the angular acceleration

The angular displacement is given by \(\theta = 3t^{2}+2t\). The angular velocity \(\omega=\frac{d\theta}{dt}=6t + 2\). The angular acceleration \(\alpha=\frac{d\omega}{dt}=6\ rad/s^{2}\)

Step2: Calculate the tangential acceleration

The formula for tangential acceleration is \(a_{t}=r\alpha\). Given \(r = 2m\) and \(\alpha=6\ rad/s^{2}\), then \(a_{t}=2\times6=12m/s^{2}\)

Step1: Find the angular velocity at \(t = 4s\)

From \(\omega=6t + 2\), when \(t = 4s\), \(\omega=6\times4+ 2=26rad/s\)

Step2: Calculate the radial acceleration

The formula for radial acceleration is \(a_{r}=r\omega^{2}\). Given \(r = 2m\) and \(\omega = 26rad/s\), then \(a_{r}=2\times26^{2}=2\times676 = 1352m/s^{2}\)

Answer:

B. \(12m/s^{2}\)

Problem 9