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 53

Chapter 8, Problem 53

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.

Verified step by step guidance

Step 1: Identify the forces acting on the ball. The ball experiences three forces: the gravitational force $ F_g $, the normal force $ F_{N1} $ from the horizontal surface, and the normal force $ F_{N2} $ from the vertical surface. These forces combine to exert a net force on the ball.

Step 2: Calculate the gravitational force $ F_g $. Use the formula $ F_g = m \cdot g $, where $ m $ is the mass of the ball (30 g or 0.03 kg) and $ g $ is the acceleration due to gravity (9.8 m/s²).

Step 3: Determine the centripetal force required for the ball to move in a circular path. The centripetal force is given by $ F_c = m \cdot \omega^2 \cdot r $, where $ \omega $ is the angular velocity in radians per second and $ r $ is the radius of the circular path (half the diameter, 0.2 m). Convert the angular velocity from rpm to radians per second using $ \omega = \frac{2 \pi \cdot \, \text{rpm}}{60} $.

Step 4: Relate the centripetal force to the net force exerted by the track. The net force exerted by the track is the vector sum of $ F_{N1} $ and $ F_{N2} $, which together provide the centripetal force. Use trigonometric relationships to resolve the components of $ F_{N1} $ and $ F_{N2} $ based on the geometry of the track.

Step 5: Combine the forces to find the magnitude of the net force. Use the Pythagorean theorem to calculate the magnitude of the net force: $ F_{\text{net}} = \sqrt{F_{N1}^2 + F_{N2}^2} $. Ensure that $ F_{N1} $ and $ F_{N2} $ are consistent with the centripetal force and gravitational force acting on the ball.

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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 that acts on an object moving in a circular path, directed towards the center of the circle. It is essential for maintaining circular motion and is calculated using the formula F_c = m(v^2/r), where m is mass, v is velocity, and r is the radius of the circular path. In this scenario, the ball experiences centripetal force due to the track's normal force.

Normal Force

The normal force is the force exerted by a surface to support the weight of an object resting on it, acting perpendicular to the surface. In the context of the ball on the track, there are two components of the normal force (F_N1 and F_N2) acting on the ball, which contribute to the net force required for circular motion. Understanding how these forces interact is crucial for solving the problem.

Net Force

Net force is the vector sum of all the forces acting on an object. It determines the object's acceleration according to Newton's second law (F_net = ma). In this case, the net force acting on the ball is the combination of gravitational force (F_g) and the normal forces (F_N1 and F_N2), which together dictate the ball's motion along the curved track.

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Textbook Question

An airplane feels a lift force $\vec{L}$ perpendicular to its wings. In level flight, the lift force points straight up and is equal in magnitude to the gravitational force on the plane. When an airplane turns, it banks by tilting its wings, as seen from behind, by an angle from horizontal. This causes the lift to have a radial component, similar to a car on a banked curve. If the lift had constant magnitude, the vertical component of $\vec{L}$ would now be smaller than the gravitational force, and the plane would lose altitude while turning. However, you can assume that the pilot uses small adjustments to the plane's control surfaces so that the vertical component of $\vec{L}$ continues to balance the gravitational force throughout the turn. Find an expression for the banking angle $θ$ needed to turn in a circle of radius $r$ while flying at constant speed $v$ .

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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

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 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

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.

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?