For a long time, I have believed that most introductory physics courses and textbooks do not cover rolling motion adequately. I feel this way because so many of the important principles of mechanics must be used when analyzing the motion of a rolling body. For example, to calculate the acceleration of a ball rolling down an incline we must use Newton’s second law and the relationship between torque and angular acceleration
(I will call this the “Rotational Equation of Motion”); we must decide whether the ball is subject to kinetic 롤대리 or static friction; and we have to relate the acceleration of the center of mass to the angular acceleration around the center of mass. Also, when there is no slipping, the conservation of mechanical energy is often applied. This requires knowing how to express the kinetic energy of a system of particles and how to relate the velocity of the center of mass to the angular velocity around the center of mass. Basically, all of the important ideas are used when solving physics problems related to rolling motion.
There’s so much physics in this subject that i will devote four articles to it. The first will cover a standard physics problem solution for rolling without slipping; the second will describe how slipping affects the motion of a rolling body; the third will describe how the motion of a body rolling without slipping can be analyzed in terms of mechanical energy conservation; and the fourth will demonstrate how seemingly difficult rolling motion problems can be solved fairly easily when the basic principles are carefully applied. Now on to the first of these four articles.
The text editor does not accept many standard mathematical symbols. As a result, I have had to use some rather unusual mathematical notation in my articles. All of that notation is described in the Ezine article “Teaching Rotational Dynamics”.
Problem. A uniform solid sphere rolls down a plane inclined at an angle th without slipping. The body’s mass, radius, and moment of inertia around the center of mass are M, R, and Icm = 2(MR**2)/5, respectively. (a) What is the sphere’s acceleration? (b) What condition must the coefficient of static friction satisfy if there is no slipping?
Solution. (a) A sphere is touching the surface of the incline, which exerts a normal force N and a frictional force f on it. The weight of the sphere has components along the incline (MGsin(th)) and perpendicular to the incline (MGcos(th)). The motion of the sphere is analyzed using an inertial coordinate system whose x axis is parallel to and directed down the incline and a y axis directed normal to and away from the incline. With the help of a free-body diagram, we have
In his 1951 Farewell Speech to Congress, General Douglas MacArthur famously said, “Once war is forced upon us, there is no other alternative than to apply every available means to bring it to a swift end. War’s very object is victory, not prolonged indecision. In war there is no substitute for victory. ”
MacArthur had been dismissed by President Harry S Truman as general in charge of the United nations Command in Korea because he publicly disagreed with Truman’s strategy of a limited war which hamstrung his efforts for victory.
The best we could achieve in Korea was a stalemate, an uneasy armistice which led us to what we have today, a divided Korea and a maniacal despot in Kim Jong Il. We never learned from MacArthur’s words, as evidenced by our ignominious loss in Vietnam and now by our imminent defeat in Afghanistan.
The consequences of a defeat in the Mideast would be far greater than losing ‘Nam. As with Korea and Vietnam, the Afghan war will be lost in Washington, D. C.
The Afghan war was “forced upon us” by the 9/11 attacks orchestrated by al Qaeda and the Taliban in Afghanistan and history may be on the verge of repeating itself not on the field of battle but in the cozy confines of the White House.
President Obama has summoned the Commander of us Forces in Afghanistan General Stanley McChrystal to the executive mansion to explain critical and dismissive comments the general and his aides made of top administration and Afghan officials.
McChrystal is in a situation very similar to that of MacArthur in Korea. He wants to win the war in Afghanistan but the administration won’t allow him to. He has issued an abject apology to the president for his remarks published in Rolling Stone; he should have resigned. MacArthur apologized to no one; he didn’t believe he was wrong.
Instead of the “limited war” restraints dictated by the Truman administration MacArthur faced in Korea, McChrystal faces the contraints of the “rules of engagement” as determined by the Obama administration.
As commander-in-chief, Obama has every right to conduct the war as he sees fit but it would be more honorable, and would save countless lives, if he simply called it quits today, began withdrawing our troops, and fired McChrystal when he meets with him tomorrow.