Mass from a density practice problems
Mass from a linear density along a rod, or a radial density across a disk. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
A thin disk of radius 2 cm has density \( \displaystyle \rho(r) = 4 e^{- r} \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A rod lies along \( \displaystyle 0 \le x \le 5 \) (meters) with linear density \( \displaystyle \rho(x) = 1 + e^{- x} \) kg/m. Find its mass.
A thin disk of radius 3 cm has density \( \displaystyle \rho(r) = 5 e^{- r} \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A rod lies along \( \displaystyle 0 \le x \le 2 \) (meters) with linear density \( \displaystyle \rho(x) = 3 x + 2 \) kg/m. Find its mass.
A thin disk of radius 4 cm has density \( \displaystyle \rho(r) = r + 5 \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A thin disk of radius 3 cm has density \( \displaystyle \rho(r) = 2 e^{- r} \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A thin disk of radius 2 cm has density \( \displaystyle \rho(r) = 5 - r^{2} \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A thin disk of radius 3 cm has density \( \displaystyle \rho(r) = 10 - r^{2} \) g/cm² at distance \( \displaystyle r \) from its center. Find its mass.
A rod lies along \( \displaystyle 0 \le x \le 6 \) (meters) with linear density \( \displaystyle \rho(x) = x^{2} + 1 \) kg/m. Find its mass.
A rod lies along \( \displaystyle 0 \le x \le 1 \) (meters) with linear density \( \displaystyle \rho(x) = 1 + e^{- x} \) kg/m. Find its mass.