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Chapter 15: Q.76 (page 419)

A $15 c m$ long, $200 g$ rod is pivoted at one end. A $20 g$ ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Short Answer

Expert verified

Period $T = 0.487 s e c$

Step by step solution

01

Given information

Pivot length $l = 15 c m$

mass of pivot $m_{1} = 200 g = 0.2 k g$

mass of ball $m_{2} = 20 g = 0.02 k g$

02

Explanation

Moment of Inertia of rod about pivot $= \frac{m_{1} l^{2}}{3}$

Moment of Inertia of clay ball about pivot $= m_{2} l^{2}$

Combine moment of inertia of pivot:

$I = (\frac{m_{1}}{3} + m_{2}) l^{2} = (\frac{0.2}{3} + 0.02) {(0.15)}^{2} I = 0.00195 k g m^{2}$

Combined mass role="math" localid="1650048680672" $M = m_{1} + m_{2} = 0.2 + 0.02 = 0.22 k g$

Time period

$T = 2 蟺 \sqrt{\frac{I}{M g l}} = 2 蟺 \sqrt{\frac{0.00195}{0.22 脳 9.81 脳 0.15}} = 0.487 \sec$

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Most popular questions from this chapter

A $200 g$ block hangs from a spring with spring constant $10 N / m$ . At the block islocalid="1650020763199" $20 cm$ below the equilibrium point and moving upward with a speed oflocalid="1650020788839" $100 cm / s$ . What are the block's

a. Oscillation frequency?

b. Distance from equilibrium when the speed is $50 cm / s$ ?

c. Distance from equilibrium at $t = 1.0 s$ ?

A pendulum on Planet $X$ , where the value of $g$ is unknown, oscillates with a period . What is the period $T = 2 s$ of this pendulum if:

a. Its mass is doubled? Explain. Note that you do not know the value of $m, L, o r g$ , so do not assume any specific values. The required analysis involves thinking about ratios.

b. Its length is doubled $?$

c. Its oscillation amplitude is doubled $?$

A spring with spring constant $15 N / m$ hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down $6.0 cm$ and released. If the ball makes $30$ oscillations in localid="1650030754517" $20 s$ , what are its localid="1650030759248" $(a)$ mass and localid="1650030763230" $(b)$ maximum speed?

A uniform rod of mass $M$ and length $L$ swings as a pendulum on a pivot at distance $L / 4$ from one end of the rod. Find an expression for the frequency $f$ of small-angle oscillations.

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in $F I G U R E E X 15.6 ?$

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