Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 62

Chapter 2, Problem 62

A steel ball rolls across a 30-cm-wide felt pad, starting from one edge. The ball's speed has dropped to half after traveling 20 cm. Will the ball stop on the felt pad or roll off?

Verified step by step guidance

Step 1: Identify the given values and the physical principles involved. The ball's initial speed decreases to half after traveling 20 cm. The width of the felt pad is 30 cm. The problem involves deceleration due to friction, and we need to determine if the ball stops before reaching the edge of the pad.

Step 2: Use the kinematic equation for motion under constant acceleration:

v'^{2} = v^{2} + 2 a d

, where

v

is the initial velocity,

v'

is the final velocity,

a

is the acceleration, and

d

is the distance traveled. Rearrange the equation to solve for acceleration

a

using the first part of the motion (20 cm).

Step 3: Substitute the known values into the equation. The ball's speed drops to half, so

v' = 0.5 v

. The distance traveled is

d = 20

cm. Solve for

a

Step 4: Use the calculated acceleration to determine if the ball stops before reaching the edge of the felt pad. Apply the kinematic equation again for the remaining distance (

d = 30 - 20 = 10

cm) to find the final velocity

v'

at the edge of the pad.

Step 5: Analyze the result. If

v'

is zero or negative, the ball stops before reaching the edge. If

v'

is positive, the ball rolls off the pad. Interpret the physical meaning of the result based on the calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. In this scenario, the felt pad exerts a frictional force on the steel ball, which slows it down as it rolls across the surface. The amount of friction depends on the materials involved and the normal force acting on the ball. Understanding friction is crucial to determine how it affects the ball's motion and whether it will stop or continue rolling.

Kinetic Energy and Work Done

Kinetic energy is the energy possessed by an object due to its motion, calculated as 1/2 mv², where m is mass and v is velocity. As the ball rolls across the felt pad, work is done against friction, converting kinetic energy into thermal energy, which slows the ball down. Analyzing the relationship between the ball's initial kinetic energy and the work done by friction helps predict whether the ball will stop before reaching the edge of the pad.

Deceleration

Deceleration refers to the rate at which an object slows down, often expressed as a negative acceleration. In this case, the ball experiences deceleration due to the frictional force from the felt pad. By understanding the deceleration and the distance traveled, we can calculate how much the ball's speed decreases and determine if it will stop before reaching the end of the pad or if it will roll off.

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