The Simple Pendulum Quiz

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 752: This expression is of the form: 16.27 where the force constant is given by and the displacement is given by . For angles less than about , the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator. Using this equation, we can find the period of a pendulum for amplitudes less than about . For the simple pendulum:

PDF page 752: for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period for a pendulum is nearly independent of amplitude, especially if is less than about . Even simple pendulum clocks can be finely adjusted and accurate. Note the dependence of on . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.

PDF page 752: for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period for a pendulum is nearly independent of amplitude, especially if is less than about . Even simple pendulum clocks can be finely adjusted and accurate. Note the dependence of on . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 751: appreciably. The linear displacement from equilibrium is , the length of the arc. Also shown are the forces on the bob, which result in a net force of toward the equilibrium position—that is, a restoring force. Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals . (The weight has components along the string and tangent to the arc.) Tension in the string exactly cancels the component parallel to the string. This leaves a net restoring force back toward the equilibrium position at . Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and differ by about 1% or less at smaller angles). Thus, for angles

PDF page 752: What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s? Strategy

PDF page 753: 16.32 Discussion This method for determining can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about .

PDF page 753: Knowing can be important in geological exploration; for example, a map of over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

PDF page 753: An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of . Pendulum 2 has a bob with a mass of . Describe how the motion of the pendula will differ if the bobs are both displaced by . Solution The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum’s length) and by the acceleration due to gravity.