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Chapter 10: Problem 1

A good baseball pitcher can throw a baseball toward home plate at \(85 \mathrm{mi} / \mathrm{h}\) with a spin of 1800 rev/min. How many revolutions does the baseball make on its way to home plate? For simplicity, assume that the \(60 \mathrm{ft}\) path is a straight line.

Short Answer

Expert verified
The baseball makes approximately 14.43 revolutions on its way to home plate.

Step by step solution

01

Convert Speed to Feet per Second

The first step is to convert the speed of the baseball from miles per hour (mi/h) to feet per second (ft/s). We know that 1 mile is equal to 5280 feet and 1 hour is equal to 3600 seconds. So, we convert 85 mi/h:\[85 \text{ mi/h} = 85 \times \frac{5280 \text{ ft}}{3600 \text{ s}}\]This simplifies to:\[85 \times \frac{5280}{3600} \approx 124.67 \text{ ft/s}\]
02

Determine Time to Reach Home Plate

Next, calculate the time it takes for the baseball to reach home plate. With a distance of 60 ft and a speed of 124.67 ft/s, the time is given by the formula:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{60 \text{ ft}}{124.67 \text{ ft/s}}\]This simplifies to:\[\text{Time} \approx 0.481 \text{ seconds}\]
03

Convert Spin to Revolutions per Second

Convert the spin rate from revolutions per minute (rev/min) to revolutions per second (rev/s). Since 1 minute is equal to 60 seconds, we have:\[1800 \text{ rev/min} = \frac{1800}{60} \text{ rev/s} = 30 \text{ rev/s}\]
04

Calculate Total Revolutions

Finally, calculate the total number of revolutions the baseball makes during its flight. Using the revolutions per second and the time in seconds, we calculate the total revolutions:\[\text{Total Revolutions} = 30 \text{ rev/s} \times 0.481 \text{ s} \approx 14.43 \text{ revolutions}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Kinematics
Rotational kinematics deals with the motion of objects that rotate. Unlike linear motion, rotational kinematics studies how objects spin or revolve around an axis. It involves parameters such as angular velocity, angular acceleration, and time.

In the case of the baseball thrown by a pitcher, we are particularly interested in its spin. When the pitcher throws the ball, it not only travels towards home plate but also spins. The key concepts here are:
  • **Angular Velocity**: This measures how quickly the object rotates. For the baseball, it was given in revolutions per minute (rev/min).
  • **Total Revolutions**: This is what how many times the ball spins as it travels to home plate.
This kind of movement can be analyzed using the principles of rotational kinematics, allowing us to determine how many complete rotations a spinning object will make over a given distance.
Understanding rotational kinematics helps us explore situations where both linear and rotational motions occur together, such as in this baseball scenario.
Conversion of Units
Conversion of units is a critical concept in solving physics problems as it ensures the calculations are in a consistent set of units. This is particularly important to avoid errors.

In the baseball problem, two main conversions took place:
  • **Speed Conversion**: Converting the baseball's speed from miles per hour (mi/h) to feet per second (ft/s) using the knowledge that 1 mile equals 5280 feet and 1 hour equals 3600 seconds. Doing this ensured that the speed and the distance the ball travels (in feet) were in compatible units.
  • **Spin Rate Conversion**: Converting the spin rate from revolutions per minute (rev/min) to revolutions per second (rev/s). We divided by 60 since there are 60 seconds in one minute.
These conversions are essential because they allow us to use formulas accurately when the variables are expressed in similar units of measurement.
Effectively managing unit conversion is a foundational skill that underpins problem-solving in physics.
Physics Problem Solving
Physics problem-solving involves a systematic process to tackle questions by applying laws and principles of physics. This often includes breaking down the problem into steps and using mathematical equations correctly.

In solving the exercise about the baseball, the process followed these steps:
  • **Understanding the Problem**: Recognizing the given data and identifying what is being asked鈥攆or example, how many revolutions the baseball makes.
  • **Converting Units**: Consistently converting measurements to ensure all variables are in compatible units before proceeding with calculations.
  • **Using Formulas**: Applying the correct formulas to derive required values such as time or revolutions.
  • **Calculating Precisely**: Ensuring accurate and step-by-step computations to avoid mistakes.
This structured approach is crucial in physics as it helps simplify complex problems, providing clarity and a pathway to reach the correct answer.

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Most popular questions from this chapter

An astronaut is tested in a centrifuge with radius \(10 \mathrm{~m}\) and rotating according to \(\theta=0.30 t^{2}\). At \(t=5.0 \mathrm{~s}\), what are the magnitudes of the (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

The angular acceleration of a wheel is \(\alpha=6.0 t^{4}-4.0 t^{2}\), with \(\alpha\) in radians per second-squared and \(t\) in seconds. At time \(t=0\), the wheel has an angular velocity of \(+2.0 \mathrm{rad} / \mathrm{s}\) and an angular position of \(+1.0\) rad. Write expressions for (a) the angular velocity (rad/s) and (b) the angular position (rad) as functions of time (s).

A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at \(10 \mathrm{rev} / \mathrm{s} ; 60\) revolutions later, its angular speed is 15 rev/s. Calculate (a) the angular acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the \(10 \mathrm{rev} / \mathrm{s}\) angular speed, and (d) the number of revolutions from rest until the time the disk reaches the 10 rev/s angular speed.

A pulley wheel that is \(8.0 \mathrm{~cm}\) in diameter has a \(5.6\) -m-long cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of \(1.5 \mathrm{rad} / \mathrm{s}^{2}\). (a) Through what angle must the wheel turn for the cord to unwind completely? (b) How long will this take?

Attached to each end of a thin steel rod of length \(1.20 \mathrm{~m}\) and mass \(6.40 \mathrm{~kg}\) is a small ball of mass \(1.06 \mathrm{~kg} .\) The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant, it is rotating at \(39.0 \mathrm{rev} / \mathrm{s}\). Because of friction, it slows to a stop in \(32.0\) s Assuming a constant retarding torque due to friction, compute (a) the angular acceleration, (b) the retarding torque, (c) the total energy transferred from mechanical energy to thermal energy by friction, and (d) the number of revolutions rotated during the \(32.0 \mathrm{~s}\) (e) Now suppose that the retarding torque is known not to be constant. If any of the quantities (a), (b), (c), and (d) can still be computed without additional information, give its value.
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