(a) Figure P9.36 shows three points in the operation of the ballistic pendulum discussed in Example 9.6 (and shown in Fig. 9.10b). The projectile approaches the pendulum in Figure P9.36a. Figure P9.36b shows the situation just after the projectile is captured in the pendulum. In Figure P9.36c, the pendulum arm has swung upward and come to rest momentarily at a height A above its initial position. Prove that the ratio of the kinetic energy of the projectile–pendulum system immediately after the collision to the kinetic energy immediately before is m 1| /( m 1 + m 2 ). (b) What is the ratio of the momentum of the system immediately after the collision to the momentum immediately before? (c) A student believes that such a large decrease in mechanical energy must be accompanied by at least a small decrease in momentum. How would you convince this student of the truth? Figure P9.36 Problem. 36 and 43. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
(a) Figure P9.36 shows three points in the operation of the ballistic pendulum discussed in Example 9.6 (and shown in Fig. 9.10b). The projectile approaches the pendulum in Figure P9.36a. Figure P9.36b shows the situation just after the projectile is captured in the pendulum. In Figure P9.36c, the pendulum arm has swung upward and come to rest momentarily at a height A above its initial position. Prove that the ratio of the kinetic energy of the projectile–pendulum system immediately after the collision to the kinetic energy immediately before is m 1| /( m 1 + m 2 ). (b) What is the ratio of the momentum of the system immediately after the collision to the momentum immediately before? (c) A student believes that such a large decrease in mechanical energy must be accompanied by at least a small decrease in momentum. How would you convince this student of the truth? Figure P9.36 Problem. 36 and 43. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
(a) Figure P9.36 shows three points in the operation of the ballistic pendulum discussed in Example 9.6 (and shown in Fig. 9.10b). The projectile approaches the pendulum in Figure P9.36a. Figure P9.36b shows the situation just after the projectile is captured in the pendulum. In Figure P9.36c, the pendulum arm has swung upward and come to rest momentarily at a height A above its initial position. Prove that the ratio of the kinetic energy of the projectile–pendulum system immediately after the collision to the kinetic energy immediately before is m1|/(m1 + m2). (b) What is the ratio of the momentum of the system immediately after the collision to the momentum immediately before? (c) A student believes that such a large decrease in mechanical energy must be accompanied by at least a small decrease in momentum. How would you convince this student of the truth?
Figure P9.36 Problem. 36 and 43. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
64. Two springs have the same unstretched length but different spring
constants, k₁ and k₂. (a) If they're connected side by side and
stretched a distance x, as shown in Fig. 4.24a, show that the force
exerted by the combination is (k₁ + k₂)x. (b) If they're con-
nected end to end (Fig. 4.24b) and the combination is stretched a
distance x, show that they exert a force k₁k2x/(k₁ + k₂).
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(a)
FIGURE 4.24 Problem 65
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(b)
65. Although we usually write Newton's second law for one-dimensional
motion in the form F =ma, which holds when mass is constant,
d(mv)
a more fundamental version is F
=
.
Consider an object
dt
whose mass is changing, and use the product rule for derivatives to
show that Newton's law then takes the form F
dm
= ma + v
dt
If a proton is located on the x-axis in some coordinate system at x0 = -3.2 x 10-5 meters, what is the x-component of the Electric Field due to this proton at a position x = +3.2 x 10-5 meters and on the x axis as the y-axis is 0 giving a number of Newtons/Coulomb?
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