1.Suppose that a simple pendulum consists of a small 75 g bob at the end of a co

Question

1.Suppose that a simple pendulum consists of a small 75 g bob at the end of a cord of negligible mass. If the angle ? between the cord and the vertical is given by
? = (0.071 rad) cos[(3.0 rad/s) t + ?],
what is its maximum kinetic energy?

2.The figure shows that if we hang a block on the end of a spring with spring constant k, the spring is stretched by distance h = 2.8 cm. If we pull down on the block a short distance and then release it, it oscillates vertically with a certain frequency. What length must a simple pendulum have to swing with that frequency?

3.The figure shows the transverse velocity u versus time t of the point on a string at x = 0, as a wave passes through it. The scale on the vertical axis is set by us = 5 m/s. The wave has form y(x, t) = ymsin(kx - ?t + ?). What is ? , in the range from - ? to +? ? (Caution: A calculator does not always give the proper inverse trig function, so check your answer by substituting it and an assumed value of ? into y(x, t) and then plotting the function.)

Explanation / Answer

given

= (0.071 rad) cos[(3.0 rad/s) t + ],

diffrentiate with respect to t

d/dt = -0.071 ( 3 ) sin ( 3t+)

again differentiate with respect to t

alpha = d^2/dt^2 =  -0.071 ( 3 )^2 cos ( 3t+) = -(3)^2

alpha = - mg L/I *

-(3)^2 = mg L/ mL^2 *

L = 9.8/ 9 = 1.088 m

the maximum kinetic energy is

K= 1/2 mw^2 xcm^2 = 1/2 mw^2 ( L thtea)^2

= 1/2 * (3)^2 ( 0.075) ( 1.088*0.071)^2

=2.01 * 10^-3 J

As per guidelines I worked first problem

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