13.1 Hookersquo;s Law

One of the simplest types of vibrational motion is that of an object attached to a spring, previously discussed in the context of energy in Chapter 5. We assume the object moves on a frictionless horizontal surface. If the spring is stretched or com-pressed a small distance from its unstretched or equilibrium position and then released, it exerts a force on the object as shown in Active Figure 13.1.From experiment the spring force is found to obey the equation

[13.1]

where x is the displacement of the object from its equilibrium position ( =0) and is a positive constant called the spring constant. This force law for springs was discovered by Robert Hooke in 1678 and is known as Hookersquo;s law. The value of is a measure of the stiffness of the spring. Stiff springs have large values, and soft springs have small values.

The negative sign in Equation 13.1 means that the force exerted by the spring is always directed opposite the displacement of the object. When the object is to the right of the equilibrium position, as in Active Figure 13.1a, is positive and is negative. This means that the force is in the negative direction, to the left. When the object is to the left of the equilibrium position, as in Active Figure 13.1c, is negative and is positive, indicating that the direction of the force is to the right. Of course, when =0, as in Active Figure 13.1b, the spring is unstretched and =0. Because the spring force always acts toward the equilibrium position, it is sometimes called a restoring force. A restoring force always pushes or pulls the object toward the equilibrium position.

Suppose the object is initially pulled a distance A to the right and released from est. The force exerted by the spring on the object pulls it back toward the equilibrium position. As the object moves toward =0, the magnitude of the force decreases (because x decreases) and reaches zero at =0. The object gains speed as it moves toward the equilibrium position, however, reaching its maximum speed when= 0. The momentum gained by the object causes it to overshoot the equilibrium position and compress the spring. As the object moves to the left of the equilibrium position (negative x-values), the spring force acts on it to the right, steadily increasing in strength, and the speed of the object decreases. The object finally comes briefly to rest at =-A before accelerating back towards= 0 and ultimately returning to the original position at= A. The process is then repeated, and the object continues to oscillate back and forth over the same path. This type of motion is called simple harmonic motion. Simple harmonic motion occurs when the net force along the direction of motion obeys Hookersquo;s law—when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.

Not all periodic motions over the same path can be classified as simple harmonic motion. A ball being tossed back and forth between a parent and a child moves repetitively, but the motion isnrsquo;t simple harmonic motion because the force acting on the ball doesnrsquo;t take the form of Hookersquo;s law, Equation 13.1. The motion of an object suspended from a vertical spring is also simple harmonic. In this case the force of gravity acting on the attached object stretches the spring until equilibrium is reached and the object is suspended at rest. By definition, the equilibrium position of the object is =0. When the object is moved away from equilibrium by a distance x and released, a net force acts toward the equilibrium position. Because the net force is proportional to , the motion is simple harmonic.

The following three concepts are important in discussing any kind of periodic motion:

■ The amplitude A is the maximum distance of the object from its equilibrium position. In the absence of friction, an object in simple harmonic motion oscillates between the positions = -A and = A.

■ The period T is the time it takes the object to move through one complete cycle of motion, from = A to = -A and back to = A.

■ The frequency f is the number of complete cycles or vibrations per unit of time, and is the reciprocal of the period ().

The acceleration of an object moving with simple harmonic motion can be found by using Hookersquo;s law in the equation for Newtonrsquo;s second law, . This gives

[13.2]

Equation 13.2, an example of a harmonic oscillator equation, gives the acceleration as a function of position. Because the maximum value of is defined to be the amplitude A, the acceleration ranges over the values - to . In the next section we will find equations for velocity as a function of position and for position as a function of time. Springs satisfying Hookersquo;s law are also called ideal springs. In real springs, spring mass, internal friction, and varying elasticity affect the force law and motion.

13.2 Elastic Potential Energy

In this section we review the material covered in Section 4 of Chapter 5.A system of interacting objects has potential energy associated with the configuration of the system. A compressed spring has potential energy that, when allowed to expand, can do work on an object, transforming spring potential energy into the objectrsquo;s kinetic energy. As an example, Figure 13.3 shows a ball being projected from a spring-loaded toy gun, where the spring is compressed a distance. As the gun is fired, the compressed spring does work on the ball and imparts kinetic energy to it.

Recall that the energy stored in a stretched or compressed spring or some otherelastic material is called elastic potential energy, , given by

[13.3]

Recall also that the law of conservation of energy, including both gravitational and spring potential energy, is given by

[13.4]

If nonconservative forces such

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13.1 Hookersquo;s Law

One of the simplest types of vibrational motion is that of an object attached to a spring, previously discussed in the context of energy in Chapter 5. We assume the object moves on a frictionless horizontal surface. If the spring is stretched or com-pressed a small distance from its unstretched or equilibrium position and then released, it exerts a force on the object as shown in Active Figure 13.1.From experiment the spring force is found to obey the equation

[13.1]

where x is the displacement of the object from its equilibrium position ( =0) and is a positive constant called the spring constant. This force law for springs was discovered by Robert Hooke in 1678 and is known as Hookersquo;s law. The value of is a measure of the stiffness of the spring. Stiff springs have large values, and soft springs have small values.

The negative sign in Equation 13.1 means that the force exerted by the spring is always directed opposite the displacement of the object. When the object is to the right of the equilibrium position, as in Active Figure 13.1a, is positive and is negative. This means that the force is in the negative direction, to the left. When the object is to the left of the equilibrium position, as in Active Figure 13.1c, is negative and is positive, indicating that the direction of the force is to the right. Of course, when =0, as in Active Figure 13.1b, the spring is unstretched and =0. Because the spring force always acts toward the equilibrium position, it is sometimes called a restoring force. A restoring force always pushes or pulls the object toward the equilibrium position.

Suppose the object is initially pulled a distance A to the right and released from est. The force exerted by the spring on the object pulls it back toward the equilibrium position. As the object moves toward =0, the magnitude of the force decreases (because x decreases) and reaches zero at =0. The object gains speed as it moves toward the equilibrium position, however, reaching its maximum speed when= 0. The momentum gained by the object causes it to overshoot the equilibrium position and compress the spring. As the object moves to the left of the equilibrium position (negative x-values), the spring force acts on it to the right, steadily increasing in strength, and the speed of the object decreases. The object finally comes briefly to rest at =-A before accelerating back towards= 0 and ultimately returning to the original position at= A. The process is then repeated, and the object continues to oscillate back and forth over the same path. This type of motion is called simple harmonic motion. Simple harmonic motion occurs when the net force along the direction of motion obeys Hookersquo;s law—when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.

Not all periodic motions over the same path can be classified as simple harmonic motion. A ball being tossed back and forth between a parent and a child moves repetitively, but the motion isnrsquo;t simple harmonic motion because the force acting on the ball doesnrsquo;t take the form of Hookersquo;s law, Equation 13.1. The motion of an object suspended from a vertical spring is also simple harmonic. In this case the force of gravity acting on the attached object stretches the spring until equilibrium is reached and the object is suspended at rest. By definition, the equilibrium position of the object is =0. When the object is moved away from equilibrium by a distance x and released, a net force acts toward the equilibrium position. Because the net force is proportional to , the motion is simple harmonic.

The following three concepts are important in discussing any kind of periodic motion:

■ The amplitude A is the maximum distance of the object from its equilibrium position. In the absence of friction, an object in simple harmonic motion oscillates between the positions = -A and = A.

■ The period T is the time it takes the object to move through one complete cycle of motion, from = A to = -A and back to = A.

■ The frequency f is the number of complete cycles or vibrations per unit of time, and is the reciprocal of the period ().

The acceleration of an object moving with simple harmonic motion can be found by using Hookersquo;s law in the equation for Newtonrsquo;s second law, . This gives

[13.2]

Equation 13.2, an example of a harmonic oscillator equation, gives the acceleration as a function of position. Because the maximum value of is defined to be the amplitude A, the acceleration ranges over the values - to . In the next section we will find equations for velocity as a function of position and for position as a function of time. Springs satisfying Hookersquo;s law are also called ideal springs. In real springs, spring mass, internal friction, and varying elasticity affect the force law and motion.

13.2 Elastic Potential Energy

In this section we review the material covered in Section 4 of Chapter 5.A system of interacting objects has potential energy associated with the configuration of the system. A compressed spring has potential energy that, when allowed to expand, can do work on an object, transforming spring potential energy into the objectrsquo;s kinetic energy. As an example, Figure 13.3 shows a ball being projected from a spring-loaded toy gun, where the spring is compressed a distance. As the gun is fired, the compressed spring does work on the ball and imparts kinetic energy to it.

Recall that the energy stored in a stretched or compressed spring or some otherelastic material is called elastic potential energy, , given by

[13.3]

Recall also that the law of conservation of energy, including both gravitational and spring potential energy, is given by

[13.4]

If nonconservative forces such

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