A child of mass 52 kg jumps onto the edge of a merry-go-round of mass 170 kg and radius 1.9 m that is initially at rest as sketched in the figure below. While in the air (during her jump), the child's linear velocity in the direction tangent to the edge of the merry-go-round is 8 m/s. What is the angular velocity of the merry-go-round plus child after the child jumps onto the merry-go-round?
To solve this problem, we can use the principle of conservation of angular momentum. Angular momentum is given by the equation:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, the merry-go-round is at rest and the child is not on it, so the total angular momentum is zero.
When the child jumps onto the merry-go-round, two things happen: the child gains angular momentum and the merry-go-round gains angular momentum.
The angular momentum gained by the child can be calculated as follows:
Lchild = mchild * vchild * r
Where mchild is the mass of the child, vchild is the linear velocity of the child, and r is the radius of the merry-go-round.
Substituting the given values, we get:
Lchild = (52 kg) * (8 m/s) * (1.9 m)
Next, let's calculate the angular momentum gained by the merry-go-round. Since it is a solid disk, its moment of inertia can be calculated using the formula:
I = (1/2) * m * r^2
Where m is the mass of the merry-go-round and r is the radius.
Substituting the given values, we get:
I = (1/2) * (170 kg) * (1.9 m)^2
Now, we have the angular momentum gained by the child and the merry-go-round. Since angular momentum is conserved, the total angular momentum after the child jumps onto the merry-go-round is equal to the sum of these two angular momenta.
Therefore,
Ltotal = Lchild + Lmerry-go-round
Setting this equal to zero, we can solve for the angular velocity ω of the merry-go-round plus child:
Itotal * ω = Lchild + Lmerry-go-round
Simplifying the equation, we get:
(1/2) * (170 kg) * (1.9 m)^2 * ω = (52 kg) * (8 m/s) * (1.9 m) + Lmerry-go-round
Finally, solving for ω, we can find the angular velocity of the merry-go-round plus child.