A child swings from a rope of length L. If she starts her swing with the rope horizontal what is her acceleration at the bottom of the swing?
a. 2gL^2
b. g/2
c. g
d. 2g/l
e. 2g
ke at bottom=PE at top
1/2 m v^2=mgh
1/2 v^2=gL
v^2=2gL
but acceleration=v^2/r=2g
To find the acceleration at the bottom of the swing, we can use the centripetal acceleration formula.
Centripetal acceleration, denoted by "a_c", is given by the formula:
a_c = (v^2) / r
Where:
- "v" is the velocity of the swinging object
- "r" is the radius of the circular motion
In this case, the swinging object is the child, and the radius of the circular motion is the length of the rope, L.
To find the velocity of the child at the bottom of the swing, we can use the principle of conservation of energy. The total mechanical energy of the child is conserved throughout the swing, so the potential energy at the highest point of the swing is converted into kinetic energy at the lowest point of the swing.
At the highest point of the swing, the potential energy is given by:
PE = mgh
Where:
- "m" is the mass of the child
- "g" is the acceleration due to gravity
- "h" is the maximum height the child reaches
The kinetic energy at the lowest point of the swing is given by:
KE = (1/2)mv^2
Where:
- "v" is the velocity of the child at the bottom of the swing
Since the total mechanical energy is conserved, we can equate the potential energy to the kinetic energy:
PE = KE
mgh = (1/2)mv^2
Canceling out the mass "m" on both sides of the equation, we get:
gh = (1/2)v^2
Rearranging the equation to solve for "v", we get:
v^2 = 2gh
Taking the square root of both sides to solve for "v", we get:
v = √(2gh)
Now that we have the velocity, we can substitute it into the centripetal acceleration formula:
a_c = (v^2) / r
a_c = (2gh) / L
Since g and L are constants, we can combine them:
a_c = (2g / L) * h
At the bottom of the swing, the maximum height "h" is equal to zero. Therefore, the acceleration at the bottom of the swing is:
a_c = (2g / L) * 0
a_c = 0
So, the correct answer is c. g. The acceleration at the bottom of the swing is equal to the acceleration due to gravity, g.