PSC # 9 CPS Physics 2012-2013

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CIRCULAR MOTION

(Centripetal Acceleration, Energy in Rotational Motion of Rigid Bodies)

Gravitational constant G = 6. 67 × 10^-11 Nm²kg^-2

Sun-Earth distance d_SE = 1. 496× 10¹¹m

Radius of the Earth R_E = 6370 km

Mass of the Earth M_E = 5. 97 × 10²⁴ kg

1. A Hummer and a Lada are making a turn. The Hummer is four times more massive than the Lada. If they make the turn at the same speed, then how do the centripetal forces acting upon the two cars compare. Explain.

2. One aspect of training for an astronaut involves experiencing and enduring extreme accelerations. To achieve this a device that rotates in a circle was designed. The astronaut is strapped into a chair in a spherical chamber. This chamber is attached to a motor in the centre of a room by a metallic, multiple poled structure I will call the arm. If the arm is 4. 50 m long and the astronaut is to experience an acceleration of 5g, calculate the following.

a) What is the linear speed of the spherical chamber?

b) What frequency (in rpm’s) must the motor generate to achieve this acceleration?

3. A flat curve on a highway has a radius of 220. 0 m. A car rounds the curve at a speed of 25. 0 ms^-1.

a) What is the minimum coefficient of friction that will prevent sliding?

b) Suppose the highway is Icy and the coefficient of friction between the tires and the pavement is only one third of what you found in part (a), what should be the maximum speed of the car so it can round the curve safely?

4. The “Giant swing “at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end (see figure). Each arm supports a seat suspended from a cable 5. 00 m long, the upper end of the cable being fastened to the arm at a point 3. 00 m from the central shaft.

(a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of

30. 0 ° with the vertical.

(b) Does the angle depend on the weight of the passenger for a given rate of revolution?

5. In another version of the “Giant swing”, the seat is connected to two cables as shown in fig, one of which is horizontal. The seat swings in a horizontal circle at a rate of 32. 0 rpm. If the seat weighs 255 N and a 825 N person is sitting in it, find the tension in each cable.

6. Assuming that the Earth is in uniform circular motion around the Sun, calculate:

a) the speed of the Earth along its trajectory

b) the magnitude of the Earth’s acceleration vector

c) the gravitational force exerted by the Earth on the Sun

7. Assume that the Earth is a perfect sphere and that the acceleration due to gravity, g, is constant at the surface and equals 9. 81 ms^-2. Neglecting the effect of the Earth’s atmosphere, calculate the horizontal speed required to place an object in circular orbit just above the surface of the Earth.

8. A geostationary satellite is a satellite directly above the Earth’s equator with a period equal to the Earth’s rotational period. Calculate the height of a geostationary satellite above the Earth’s surface.

9. Consider a conical pendulum consisting of a bob of mass m at the end of a massless cord of length L. The bob moves in a horizontal circle with constant speed v, with the wire making a constant angle β with the vertical direction.

a) Draw a free body diagram for the bob

b) Find an expression for the tension T in the cord as a function of m, g, and β .

c) Find an expression for the period T of the pendulum (the time for one revolution of the bob) as a function of L, g, and β.

10. Uniform circular motion in a vertical circle

A passenger on a big wheel moves in a vertical circle of radius R with constant speed v. During the motion, the passenger’s seat remains upright. Find an expression for the force that the seat exerts on the passenger (a) at the top of the circle and (b) at the bottom of the circle.

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