10. Conservation of Energy

A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius R . From what minimum height h must the block start to make it around without falling off? Give your answer as a multiple of R.

A 50 g mass is attached to a light, rigid, 75-cm-long rod. The other end of the rod is pivoted so that the mass can rotate in a vertical circle. What speed does the mass need at the bottom of the circle to barely make it over the top of the circle?

How much work is done by the environment in the process shown in FIGURE EX10.39? Is energy transferred from the environment to the system or from the system to the environment?

A 62-kg trampoline artist jumps upward from the top of a platform with a vertical speed of 4.5 m/s. If the trampoline behaves like a spring of spring constant 5.8 x 10⁴ N/m, how far does he depress it?

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Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the velocity of the block at point B.

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the thermal energy produced as the block slides from B to C.

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the velocity of the block at point C.

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the stiffness constant k for the spring.

In one day, a $75$-kg mountain climber ascends from the $1500$-m level on a vertical cliff to the top at $2400$ m. The next day, she descends from the top to the base of the cliff, which is at an elevation of $1350$ m. What is her change in gravitational potential energy on the first day?

A 12 kg weather rocket generates a thrust of 200 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 550 N/m, is anchored to the ground. After the engine is ignited, what is the rocket’s speed when the spring has stretched 40 cm?

Sketch the n = 8 wave function for the potential energy shown in FIGURE EX40.13.

In which of the following reactions or decays is strangeness conserved? In each case, explain your reasoning.

(a) $K^{+}+{\mu }^{+}+{\nu }_{\mu }$

(b) $n+K^{+}\to p+{\pi }^0$

(c) $K^{+}+K^{-}\to {\pi }^0+{\pi }^0$

(d) $p+K^{-}\to {\Lambda }^0+{\pi }^0$