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Chapter 8: Problem 62

A ball of radius 0.200 m rolls with a constant linear speed of \(3.60 \mathrm{m} / \mathrm{s}\) along a horizontal table. The ball rolls off the edge and falls a vertical distance of \(2.10 \mathrm{m}\) before hitting the floor. What is the angular displacement of the ball while the ball is in the air?

Short Answer

Expert verified
The angular displacement is 11.75 radians.

Step by step solution

01

Determine Time of Fall

First, we need to find out how long the ball is in the air. We use the vertical motion equation for free fall: \[ h = \frac{1}{2}gt^2 \]where \( h = 2.10 \text{ m} \) and \( g = 9.81 \text{ m/s}^2 \). Solving for \( t \):\[ t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 2.10}{9.81}} \approx 0.654 \text{ seconds} \].
02

Calculate Linear Distance Traveled

Now we calculate how far the ball travels horizontally during the fall. Use the formula:\[ \text{distance} = \text{speed} \times \text{time} \]where the linear speed is \(3.60 \text{ m/s}\) and time is \(0.654 \text{ seconds}\). Thus,\[ \text{distance} = 3.60 \times 0.654 = 2.35 \text{ meters} \].
03

Relate Linear and Angular Displacement

With the distance traveled known, we can find the angular displacement. Recall that the linear distance \( s \) traveled by the ball's edge is related to its angular displacement \( \theta \) via:\[ s = r \theta \]where \( r = 0.200 \text{ meters} \) is the radius. Thus,\[ \theta = \frac{s}{r} = \frac{2.35}{0.200} = 11.75 \text{ radians} \].

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Key Concepts

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

Angular Displacement
Angular displacement is a measure of how much an object has rotated. It's like counting how many circles a ball rolls through along its path. For our ball, the angular displacement is how many radians it rotates while falling off the table.
  • To find angular displacement, we use the formula: \[ \theta = \frac{s}{r} \]
  • Where \( \theta \) is the angular displacement in radians, \( s \) is the linear distance traveled, and \( r \) is the radius of the circle.
In this case, the ball travels 2.35 meters with a 0.200-meter radius. Therefore, the ball rotates 11.75 radians during the fall. Radia refer to slices of the circle much like the slices of a pie, explaining how far it turns.
Linear Speed
Linear speed is the rate at which an object moves along a path. For our rolling ball, it tells us how fast it travels across the table and during its fall.
  • This speed is measured in meters per second (m/s), indicating the distance the ball covers in one second.
  • In our scenario, the ball rolls at a constant linear speed of 3.60 m/s.
Linear speed helps us determine how far the ball moves in a certain time. While the ball is in the air, the same speed continues horizontally, unaffected by gravity. This aids in calculating the total distance it rolls as it falls.
Free Fall
Free fall occurs when gravity is the only force acting on the object, like our ball after it leaves the table edge. In free fall, objects accelerate due to gravity without other pushes or pulls.
  • The acceleration due to gravity is approximately 9.81 m/s虏 on Earth鈥檚 surface.
  • In this exercise, the ball falls vertically 2.10 meters purely under gravity's influence.
Even though the ball travels horizontally at constant speed, its vertical speed increases due to gravity. By calculating the time the ball takes to fall, we utilize this concept to predict how it moves.
Vertical Motion Equation
The vertical motion equation is a crucial tool in predicting how objects fall under gravity. We use the equation:
\[ h = \frac{1}{2}gt^2 \]
  • Here, \( h \) is the vertical distance, \( g \) is the gravitational acceleration (9.81 m/s虏), and \( t \) is the fall time.
This equation helps find how long the ball remains in the air, based on the height it falls from (2.10 meters in our case). Solving for \( t \), we learn that the ball takes approximately 0.654 seconds to hit the ground. Knowing this time is key for determining other factors like horizontal distance traveled and angular displacement of the ball.

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Most popular questions from this chapter

A wind turbine is initially spinning at a constant angular speed. As the wind's strength gradually increases, the turbine experiences a constant angular acceleration of \(0.140 \mathrm{rad} / \mathrm{s}^{2} .\) After making 2870 revolutions, its angular speed is \(137 \mathrm{rad} / \mathrm{s}\). (a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

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 "ball," 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.

The penny-farthing is a bicycle that was popular between 1870 and \(1890 .\) As the drawing shows, this type of bicycle has a large front wheel and a small rear wheel. During a ride, the front wheel (radius \(=1.20 \mathrm{m})\) makes 276 revolutions. How many revolutions does the rear wheel (radius \(=0.340 \mathrm{m}\) ) make?

A stroboscope is a light that flashes on and off at a constant rate. It can be used to illuminate a rotating object, and if the flashing rate is adjusted properly, the object can be made to appear stationary. (a) What is the shortest time between flashes of light that will make a three-bladed propeller appear stationary when it is rotating with an angular speed of \(16.7 \mathrm{rev} / \mathrm{s} ?\) (b) What is the next shortest time?

Three Wheels. Three rubber wheels are mounted on axles so that their outer edges make tight contact with each other and their centers are on a line. The wheel on the far left axle is connected to a motor that rotates it at \(25.0 \mathrm{rpm},\) and drives the wheel in contact with it on its right, which, in turn, drives the wheel on its right. The left wheel (Wheel 1) has a diameter of \(d_{1}=0.20 \mathrm{m},\) the middle wheel (Wheel \(\left.2\right)\) has \(d_{2}=0.30 \mathrm{m},\) and the far right wheel (Wheel 3) has \(d_{3}=0.45 \mathrm{m} .\) (a) If Wheel 1 rotates clockwise, in which direction does Wheel 3 rotate? (b) What is the angular speed of Wheel \(3,\) and what is the tangential speed on its outer edge? (c) What arrangement of the wheels gives the largest tangential speed on the outer edge of the wheel in the far right position (assuming the wheel in the far left position is driven at \(25.0 \mathrm{rpm}) ?\)
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