Question

1.Earth revolves around the sun with a speed of 30 km/s in an approximate circular path.

How much acceleration does it have directed towards the sun? Orbital radius of the earth

is 1.5 108 km.

2. An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15o.

What is the radius of the loop?

3. A car of mass M experiences a maximum frictional force of 0.65Mg pavement as it goes

around the curved road of radius 150m. How fast can the car be moving to successfully go

around the curve?

Answer 1

1) We can find the acceleration directed towards the Sun as follows:

$$a_c=\dfrac{v^2}{r}=\dfrac{(3\cdot10^4\ \dfrac{m}{s})^2}{1.5\cdot10^{11}\ m}=6.0\cdot10^{-3}\ \dfrac{m}{s^2}.$$

2) Let's apply Newton's Second Law of Motion in projections on axis $x$- and $y$:

$$Ncos\theta=mg,$$ $$Nsin\theta=\dfrac{mv^2}{r}.$$

Dividing the second equation by the first one, we get:

$$tan\theta=\dfrac{v^2}{rg},$$ $$r=\dfrac{v^2}{gtan\theta},$$ $$r=\dfrac{(200\ \dfrac{m}{s})^2}{9.8\ \dfrac{m}{s}\cdot tan15^{\circ}}=15.23\ km.$$

3)

$$F_c=F_{fr},$$ $$\dfrac{Mv^2}{r}=\mu Mg,$$ $$v=\sqrt{\mu rg}=\sqrt{0.65\cdot150\ m\cdot9.8\ \dfrac{m}{s^2}}=31\ \dfrac{m}{s}.$$