A certain uniform spring has spring constant. Now the spring is cut in half. What is the relationship between and the spring constant of each resulting smaller spring? Explain your reasoning.
When a spring of spring constant K is cut in half the spring constant of each of the resulting smaller springs is 2K .
A spring of spring constant K is cut in half.
Two springs of the same material but having different lengths l1 and l2 have their spring constants K1 and K2 related as
Let the initial length be L . Length of the new pieces when the spring is cut in half is L/2.
Thus from equation (I)
Hence spring constant of each of the new pieces is2 K.
Jonathan is riding a bicycle and encounters a hill of height h. At the base of the hill, he is traveling at a speed . When he reaches the top of the hill, he is traveling at a speed role="math" localid="1663652623999" . Jonathan and his bicycle together have a mass m. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathan’s body during this process? (c) How much work does Jonathan do on the bicycle pedals within the Jonathan– bicycle–Earth system during this process?
A ball of mass is connected by a strong string of length to a pivot and held in place with the string vertical. A wind exerts constant force to the right on the ball as shown in Figure P8.82. The ball is released from rest. The wind makes it swing up to attain maximum height above its starting point before it swings down again. (a) Find as a function of . Evaluate for (b) and (c) . How does behave (d) as approaches zero and (e) as approaches infinity? (f) Now consider the equilibrium height of the ball with the wind blowing. Determine it as a function of . Evaluate the equilibrium height for (g) and (h) going to infinity.
A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig. P8.68). (a) Show that if the sphere is released from a height below that of the peg. It will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position and is to swing in a complete circle centered on the peg, the minimum value of d must be .
Review: Why is the following situation impossible? A new high-speed roller coaster is claimed to be so safe that the passengers do not need to wear seat belts or any other restraining device. The coaster is designed with a vertical circular section over which the coaster travels on the inside of the circle so that the passengers are upside down for a short time interval. The radius of the circular section is 12.0 m, and the coaster enters the bottom of the circular section at a speed of 22.0 m/s. Assume the coaster moves without friction on the track and model the coaster as a particle.
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