Ch 08: Dynamics II: Motion in a Plane

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 08: Dynamics II: Motion in a Plane

Problem 52a

Chapter 8, Problem 52a

Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ω_min you must maintain if you want the ball to complete the full circle without the string going slack at the top. Find an expression for ω_min.

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Step 1: Analyze the forces acting on the ball at the top of the circle. At this point, the tension in the string is zero (since the string is just about to go slack), and the only force acting on the ball is its weight, which provides the centripetal force required to keep the ball moving in a circular path.

Step 2: Write the equation for centripetal force at the top of the circle. The centripetal force is provided entirely by the gravitational force: F_c = m * g, where m is the mass of the ball and g is the acceleration due to gravity.

Step 3: Relate the centripetal force to the angular velocity. The centripetal force can also be expressed as F_c = m * ω² * L, where ω is the angular velocity and L is the length of the string (radius of the circle).

Step 4: Set the two expressions for centripetal force equal to each other: m * g = m * ω² * L. Simplify this equation by canceling out the mass m on both sides.

Step 5: Solve for the minimum angular velocity ωₘᵢₙ. Rearrange the equation to isolate ω: ωₘᵢₙ = √(g / L). This is the expression for the minimum angular velocity required for the ball to complete the circle without the string going slack.

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Key Concepts

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

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In the case of the ball on a string, this force is provided by the tension in the string and the gravitational force acting on the ball. For the ball to complete a vertical circle, the centripetal force must be sufficient to counteract the effects of gravity, especially at the top of the circle.

Minimum Angular Velocity

Minimum angular velocity (ωₘᵢₙ) is the lowest speed at which an object must move in a circular path to maintain its motion without falling due to gravity. At the top of the vertical circle, the gravitational force must provide enough centripetal force to keep the ball in motion. If the angular velocity is below this minimum, the tension in the string becomes zero, causing the ball to fall.

Energy Conservation

Energy conservation in a vertical circular motion involves the transformation between kinetic and potential energy. As the ball moves up and down in the circle, its potential energy increases at the top and decreases at the bottom, while its kinetic energy changes inversely. This principle helps in deriving the minimum angular velocity by equating the potential energy at the top with the kinetic energy required to maintain circular motion.

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Textbook Question

The physics of circular motion sets an upper limit to the speed of human walking. (If you need to go faster, your gait changes from a walk to a run.) If you take a few steps and watch what's happening, you'll see that your body pivots in circular motion over your forward foot as you bring your rear foot forward for the next step. As you do so, the normal force of the ground on your foot decreases and your body tries to 'lift off' from the ground. A person's center of mass is very near the hips, at the top of the legs. Model a person as a particle of mass m at the top of a leg of length L. Find an expression for the person's maximum walking speed v_max.

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Textbook Question

A 2.0 kg pendulum bob swings on a 2.0-m-long string. The bob's speed is 1.5 m/s when the string makes a 15° angle with vertical and the bob is moving toward the bottom of the arc. At this instant, what are the magnitudes of the tension in the string?

Textbook Question

In an amusement park ride called The Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in FIGURE P8.51. What is the longest rotation period of the wheel that will prevent the riders from falling off at the top?

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Textbook Question

A 30 g ball rolls around a 40-cm-diameter L-shaped track, shown in FIGURE P8.53, at 60 rpm. What is the magnitude of the net force that the track exerts on the ball? Rolling friction can be neglected. Hint: The track exerts more than one force on the ball.

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Textbook Question

A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. FIGURE P8.48 shows that the string traces out the surface of a cone, hence the name. Find an expression for the ball's angular speed ω.

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Textbook Question

FIGURE P8.54 shows two small 1.0 kg masses connected by massless but rigid 1.0-m-long rods. What is the tension in the rod that connects to the pivot if the masses rotate at 30 rpm in a horizontal circle?