Question

A spring of negligible mass has force constant k=1600k = 1600 N/m. You place the spring vertically with one end on the floor. You then drop a 1.201.20-kg book onto it from a height of 0.8000.800 m above the top of the spring. Find the maximum distance the spring will be compressed.

Accepted Answer

Step 1: Identify the energy transformations involved. The book initially has gravitational potential energy due to its height above the spring. As the book falls, this energy is converted into kinetic energy, and finally, when the book compresses the spring, the energy is stored as elastic potential energy in the spring. Step 2: Write the equation for conservation of energy. The total mechanical energy is conserved, so the initial gravitational potential energy of the book is equal to the elastic potential energy stored in the spring at maximum compression. Use the formula for gravitational potential energy, $$ E_{g} = mgh $$, and elastic potential energy, $$ E_{s} = \frac{1}{2}kx^2 $$, where $$ x  is $$the compression distance. Step 3: Substitute the given values into the gravitational potential energy formula. The mass of the book $$ m  is 1.20 kg$$, the height $$ h  is 0.800 m$$, and the acceleration due to gravity $$ g  is $$approximately 9.8 m/s². Calculate $$ E_{g} . $$Step 4: Set $$ E_{g} $$ equal to $$ E_{s}  to $$find the compression distance $$ x . $$Rearrange the elastic potential energy formula $$ \frac{1}{2}kx^2 = mgh  to $$solve for $$ x . $$Substitute $$ k = 1600  N/m $$and the calculated $$ E_{g} $$ into the equation. Step 5: Solve for $$ x $$ algebraically. Take the square root of both sides after isolating $$ x^2 . $$This will give the maximum compression distance of the spring.

A spring of negligible mass has force constant $k equals 1600$ N/m. You place the spring vertically with one end on the floor. You then drop a $1.20$ -kg book onto it from a height of $0.800$ m above the top of the spring. Find the maximum distance the spring will be compressed.

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Concetti chiave

Hooke's Law

Conservation of Energy

Gravitational Potential Energy