Projectile Motion Experiment

Astronauts fall around the Earth at thousands of meters per second, held by the centripetal force provided by gravity.

Rotational motion is an important part of everyday life. The rotation of the Earth creates the cycle of day and night, the rotation of wheels enables easy vehicular motion, and modern technology depends on circular motion in a variety of contexts, from the tiny gears in a Swiss watch to the operation of lathes and other machinery. The concepts of angular speed, angular acceleration, and centripetal acceleration are central to understanding the motions of a diverse range of phenomena, from a car moving around a circular racetrack to clusters of galaxies orbiting a common center.

Rotational motion, when combined with Newton’s law of universal gravitation and his laws of motion, can also explain certain facts about space travel and satellite motion, such as where to place a satellite so it will remain fixed in position over the same spot on the Earth. The generalization of gravitational potential energy and energy conservation offers an easy route to such results as planetary escape speed. Finally, we present Kepler’s three laws of planetary motion, which formed the foundation of Newton’s approach to gravity.

7.1 ANGULAR SPEED AND ANGULAR ACCELERATION
In the study of linear motion, the important concepts are displacement Ax, velocity v, and acceleration a. Each of these concepts has its analog in rotational motion: angular displacement Δθ, angular velocity ω, and angular acceleration a.

The radian, a unit of angular measure, is essential to the understanding of these concepts. Recall that the distance s around a circle is given by s = 2πr, where r is the radius of the circle. Dividing both sides by r results in s/r = 2π. This quantity is dimensionless because both s and r have dimensions of length, but the value 2π corresponds to a displacement around a circle. A half circle would give an answer of p, a quarter circle an answer of π/2. The numbers 2π, π, and π/2 correspond to angles of 360°, 180°, and 90°, respectively, so a new unit of angular measure, the radian, can be introduced, with 180° = π rad relating degrees to radians.

The angle θ subtended by an arc length s along a circle of radius r, measured in radians counterclockwise from the positive x-axis, is

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The angle θ in Equation 7.1 is actually an angular displacement from the positive x-axis, and s the corresponding displacement along the circular arc, again measured from the positive x-axis. Figure 7.1 illustrates the size of 1 radian, which is approximately 57°. Converting from degrees to radians requires multiplying by the ratio (π rad/180°). For example, 45° (π rad/180°) = (π/4) rad.

Tip 7.1 Remember the Radian
Equation 7.1 uses angles measured in radians. Angles expressed in terms of degrees must first be converted to radians. Also, be sure to check whether your calculator is in degree or radian mode when solving problems involving rotation.

For very short time intervals, the average angular speed approaches the instantaneous angular speed, just as in the linear case.

The instantaneous angular speed ω of a rotating rigid object is the limit of the average speed Δθ/Δt as the time interval Δt approaches zero:

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SI unit: radian per second (rad/s)

We take ω to be positive when θ is increasing (counterclockwise motion) and negative when θ is decreasing (clockwise motion). When the angular speed is constant, the instantaneous angular speed is equal to the average angular speed.

EXAMPLE 7.1 Whirlybirds
Goal Convert an angular speed in revolutions per minute to radians per second.

Problem The rotor on a helicopter turns at an angular speed of 3.20 × 102 revolutions per minute. (In this book, we sometimes use the abbreviation rpm, but in most cases we use rev/min.) (a) Express this angular speed in radians per second. (b) If the rotor has a radius of 2.00 m, what arclength does the tip of the blade trace out in 3.00 × 102 s?

Strategy During one revolution, the rotor turns through an angle of 2π radians. Use this relationship as a conversion factor.

Solution

· (a) Express this angular speed in radians per second.Apply the conversion factors 1 rev = 2π rad and 60 s = 1 min:image5.jpg

· (b) Multiply the angular speed by the time to obtain the angular displacement:image6.jpgMultiply the angular displacement by the radius to get the arc length:image7.jpg

Remarks In general, it’s best to express angular speeds in radians per second. Consistent use of radian measure minimizes errors.

When a rigid object rotates about a fixed axis, as does the bicycle wheel, every portion of the object has the same angular speed and the same angular acceleration. This fact is what makes these variables so useful for describing rotational motion. In contrast, the tangential (linear) speed and acceleration of the object take different values that depend on the distance from a given point to the axis of rotation.

7.2 ROTATIONAL MOTION UNDER CONSTANT ANGULAR ACCELERATION
A number of parallels exist between the equations for rotational motion and those for linear motion. For example, compare the defining equation for the average angular speed,

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with that of the average linear speed,

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In these equations, v takes the place of v and θ takes the place of x, so the equations differ only in the names of the variables. In the same way, every linear quantity we have encountered so far has a corresponding “twin” in rotational motion.

The procedure used in Section 2.5 to develop the kinematic equations for linear motion under constant acceleration can be used to derive a similar set of equations for rotational motion under constant angular acceleration. The resulting equations of rotational kinematics, along with the corresponding equations for linear motion, are as follows:

Linear Motion with a Constant (Variables: x and v)

Rotational Motion About a Fixed Axis with a Constant (Variables: θ and ω)

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EXAMPLE 7.2 A Rotating Wheel

Goal Apply the rotational kinematic equations.

Problem A wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular speed of the wheel is 2.00 rad/s at t = 0, (a) through what angle does the wheel rotate between t = 0 and t = 2.00 s? Give your answer in radians and in revolutions. (b) What is the angular speed of the wheel at t = 2.00 s?

Strategy The angular acceleration is constant, so this problem just requires substituting given values into Equations 7.7 and 7.8.

Solution

· (a) Find the angular displacement after 2.00 s, in both radians and revolutions.Use Equation 7.8, setting ωi = 2.00 rad/s, α = 3.5 rad/s2, and t = 2.00 s:image12.jpgConvert radians to revolutions.image13.jpg

· (b) What is the angular speed of the wheel at t = 2.00 s?Substitute the same values into Equation 7.7:image14.jpg

Remarks The result of part (b) could also be obtained from Equation 7.9 and the results of part (a).

EXAMPLE 7.3 Slowing Propellers
Goal Apply the time-independent rotational kinematic equation.

Problem An airplane propeller slows from an initial angular speed of 12.5 rev/s to a final angular speed of 5.00 rev/s. During this process, the propeller rotates through 21.0 revolutions. Find the angular acceleration of the propeller in radians per second squared, assuming it’s constant.

Strategy The given quantities are the angular speeds and the displacement, which suggests applying Equation 7.9, the time-independent rotational kinematic equation, to find a.

Solution

First, convert the angular displacement to radians and the angular speeds to rad/s:

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Substitute these values into Equation 7.9 to find the angular acceleration a:

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Solve for α:

Remark Waiting until the end to convert revolutions to radians is also possible and requires only one conversion instead of three.

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The tangential acceleration of a point on a rotating object equals the distance of that point from the axis of rotation multiplied by the angular acceleration. Again, radian measure must be used for the angular acceleration term in this equation.

One last equation that relates linear quantities to angular quantities will be derived in the next section.