/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 At the local swimming hole, a fa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 31

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 \(\mathrm{m}\) above the water. One diver runs off the edge of the cliff, tucks into a 鈥渂all,鈥� and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

Short Answer

Expert verified
The diver makes approximately 2.08 revolutions.

Step by step solution

01

Determine the Time of Fall

The diver falls from a height of 8.3 meters. To find the time taken to fall, we use the kinematic equation of motion for free fall: \[s = \frac{1}{2} g t^2\]where \(s\) is the distance (8.3 m), \(g\) is the acceleration due to gravity (9.8 m/s虏), and \(t\) is the time. Solving for \(t\), we have:\[8.3 = \frac{1}{2} \times 9.8 \times t^2\]\[t^2 = \frac{16.6}{9.8}\]\[t \approx \sqrt{1.694}\]\[t \approx 1.30 \text{ seconds}\]
02

Calculate the Number of Revolutions

Now, calculate the number of revolutions by using the average angular speed. The angular speed is 1.6 revolutions per second (rev/s), and the time of fall is 1.30 seconds. The formula to find the number of revolutions is:\[\text{Number of Revolutions} = \text{Angular Speed} \times \text{Time}\]\[\text{Number of Revolutions} = 1.6 \times 1.30\]\[\text{Number of Revolutions} \approx 2.08\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Kinematic Equations
Kinematic equations are essential tools in physics for describing motion. These equations relate the five key variables of motion under constant acceleration: displacement (s), initial velocity (v_0), final velocity (v), acceleration (a), and time (t). In our scenario, the diver falls from rest, so the initial velocity is zero. The acceleration is due to gravity, which is a constant 9.8 m/s虏 on Earth. By using the specific kinematic equation for free fall, \[ \ s = \frac{1}{2} gt^2 \], we can solve for time
  • Displacement (s): 8.3 meters
  • Gravity (g): 9.8 m/s虏
  • Time (t): Unknown
For our diver's fall, we rearrange this equation to solve for time, knowing the distance she travels (8.3 m) and the acceleration (9.8 m/s虏). This allows us to derive the time of fall as approximately 1.30 seconds. These kinematic equations provide the foundation for analyzing any motion.
Angular Speed
Angular speed refers to how fast an object rotates or revolves around an axis. It's an important concept whenever you're dealing with circular motion. The angular speed is measured in revolutions per second (rev/s) or radians per second (rad/s). In the diver's case, her average angular speed is 1.6 rev/s. This means she completes 1.6 cycles of rotation every second while in midair.

To determine the number of revolutions during her fall, we can multiply the angular speed by the time of fall. With a known time of fall (1.30 seconds), we find that the diver completes about 2.08 revolutions. Understanding angular speed is crucial in determining these rotations, providing insights into dynamic movements of rotating bodies.
Free Fall
Free fall is the motion of an object under the influence of gravitational force only. During free fall, all objects experience the same acceleration due to gravity, regardless of their mass. It's an excellent example of Newton's laws of motion. In our exercise, the diver jumps off a cliff, experiencing free fall until she hits the water.

The kinematic equation for free fall, \( s = \frac{1}{2} gt^2 \), lets us calculate how long she is in the air before reaching the water. Ignoring air resistance makes it a perfect scenario for examining pure free-fall motion. From this calculation, we measured her fall time to be approximately 1.30 seconds. This understanding is vital in calculating other factors like her speed upon hitting the water or her number of rotations.
Revolutions
Revolutions refer to the number of complete turns or cycles an object makes around a central axis. This number is crucial in analyzing rotational motion. During the diver's plunge, she completes 2.08 full turns before landing in the water. The number of revolutions is calculated by multiplying the diver's angular speed, which remains constant at 1.6 rev/s, by the total time she spends falling.
  • Time in air: 1.30 seconds
  • Angular speed: 1.6 rev/s
Revolutions help describe rotational kinetics and provide a clear measure of motion for objects spinning around. It's a straightforward yet significant measurement in physics, particularly when assessing objects like wheels, planets, or divers in free fall.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The wheels of a bicycle have an angular velocity of 20.0 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of 15.92 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the angular acceleration (in rad/s2 ) of each wheel?

A flywheel has a constant angular deceleration of 2.0 \(\mathrm{rad} / \mathrm{s}^{2}\) (a) Find the angle through which the flywheel tums as it comes to rest from an angular speed of 220 \(\mathrm{rad} / \mathrm{s}\) . (b) Find the time for the flywheel to come to rest.

A thin rod (length 1.50 m) is oriented vertically, with its bottom end attached to the floor by means of a frictionless hinge. The mass of the rod may be ignored, compared to the mass of an object fixed to the top of the rod. The rod, starting from rest, tips over and rotates downward. (a) What is the angular speed of the rod just before it strikes the floor? (Hint: Consider using the principle of conservation of mechanical energy.) (b) What is the magnitude of the angular acceleration of the rod just before it strikes the floor?

ssm A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of 330 rev/min (assumed constant) on its way to the catcher鈥檚 mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

An electric drill starts from rest and rotates with a constant angular acceleration. After the drill has rotated through a certain angle, the magnitude of the centripetal acceleration of a point on the drill is twice the magnitude of the tangential acceleration. What is the angle?
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Wave Optics

Read Explanation

Mechanics and Materials

Read Explanation

Quantum Physics

Read Explanation

Electrostatics

Read Explanation

Kinematics Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.