Question

A solid brass cylinder and a solid wood cylinder have the same radius and mass (the wood cylinder is longer). Released together from rest, they roll down an incline. (a) Which cylinder reaches the bottom first, or do they tie? (b) The wood cylinder is then shortened to match the length of the brass cylinder, and the brass cylinder is drilled out along its long (central) axis to match the mass of the wood cylinder. Which cylinder now wins the race, or do they tie?

Answer 1

The solid brass cylinder will reach the bottom first, while the wood cylinder will reach the bottom at the same time.

Step-by-step explanation:

A solid brass cylinder and a solid wood cylinder with the same radius and mass, but different lengths, are released together from rest and roll down an incline. The cylinder that reaches the bottom first depends on their moments of inertia. The moment of inertia for a solid cylinder is given by I = (1/2)MR², where M is the mass and R is the radius. The moment of inertia for a solid wood cylinder will be larger due to its longer length, so the brass cylinder will reach the bottom first.

When the wood cylinder is shortened to match the length of the brass cylinder and the brass cylinder is drilled out along its long axis to match the mass of the wood cylinder, their moments of inertia become equal. In this case, both cylinders will reach the bottom at the same time and tie.

Answer 2

a. They will be tie

b. Win the wood cylinder

Step-by-step explanation:

a.

The both cylinders will reach the bottom at the same time notice the relation in the equation in indepent of the length and both have the same radius and the same rotational inertia.

`I=(1)/(2)*m*r^2`

`a=(g*sin(\beta))/(1+I_(com)/m*r^2)`

So both will be tie

b.

`a_(brass)=(g*sin(\beta))/(1+I_(brass)/m*r^2)=a_(wood)=(g*sin(\beta))/(1+I_(wood)/m*r^2)`

The acceleration of the wood cylinder is larger than the acceleration of the brass cylinder so the cylinder of wood will reach the bottom first

`a_(brass)<a_(wood)`

So the wood win the race