Gravity Exponents Bell Ringer
Algebra II • STEM Warm-Up Active Monitoring Protocol
Gravity Exponents Bell Ringer
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Law of Gravitation: \( F_g = G \frac{m_1 m_2}{r^2} \implies r = \left( \frac{G \cdot m_1 \cdot m_2}{F_g} \right)^{1/2} \)
Exponent Rules: \( 10^a \cdot 10^b = 10^{a+b} \) | \( \frac{10^a}{10^b} = 10^{a-b} \) | \( (10^a)^{1/2} = 10^{a/2} \)
1
Deconstruct & Simplify Powers of 10
Coefficient-Exponent Separation
Given data: \( G = 6.0 \times 10^{-11} \), \( m_1 = 4.0 \times 10^{24} \), \( m_2 = 9.0 \times 10^{22} \), and \( F_g = 6.0 \times 10^{19} \). Separate the numerical coefficients from powers of 10 to evaluate \( r^2 \):
A. Combine Coefficients:
Compute: \( \frac{6.0 \times 4.0 \times 9.0}{6.0} \)
B. Combine Exponent Powers:
Compute: \( \frac{10^{-11} \cdot 10^{24} \cdot 10^{22}}{10^{19}} = 10^{\text{?}} \)
Teacher Checkpoint 1 • Raise Hand Before Proceeding
Teacher checks: Numerator sum \( (-11+24+22=35) \), quotient \( (35-19=16) \), coefficient = \( 36.0 \).
Teacher Stamp:
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2
Solve for Distance Using Rational Exponents
Power of a Power Rule
From Part 1, \( r^2 = 36.0 \times 10^{16} \). Apply the rational power \( \frac{1}{2} \) to both factors: \( r = (36.0 \times 10^{16})^{1/2} = (36.0)^{1/2} \times (10^{16})^{1/2} \).
Evaluate \( (36.0)^{1/2} \):
Coefficient square root
Evaluate \( (10^{16})^{1/2} \):
Multiply: \( 16 \times \frac{1}{2} \)
Final Radius \( r \) (meters):
Scientific notation
Teacher Checkpoint 2 • Verification & Diagnostic Check
Diagnostic Trap: Look for \( 10^8 \) (correct) vs \( 10^4 \) (square rooting exponent) or \( 10^{16} \) (forgot exponent).
Mastery Stamp:
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3
Algebraic Scaling Challenge (No Calculator)
Rational Power Reasoning
An astronomer models an orbit where distance increases by a factor of \( 10^{3/2} \). By what exact factor does \( r^2 \) in the denominator increase? Show your step using the power-of-a-power rule \( (10^{3/2})^2 \), then express your final factor as a standard integer.
Teacher Quick Key: • Check 1: \( r^2 = 36.0 \times 10^{16} \) • Check 2: \( r = 6.0 \times 10^8 \text{ m} \) • Part 3: \( 10^{(3/2) \cdot 2} = 10^3 = 1{,}000 \)
Algeb-II-GRV-01
