How to find translational speed?
A bowling ball encounters a 0.760 m vertical rise on the way back to the ball rack. Ignore frictional losses and assume that the mass of the ball is distributed uniformly. The translational speed of the ball is 4.70 m/s at the bottom of the rise. Find the translational speed at the top
This is an energy problem (well, everything’s an energy problem!). You’ve got two energies being traded: the kinetic energy (energy of motion) of the ball, and the potential energy (energy of position) of the ball.
The kinetic energy is related to velocity by
KE = 1/2 * mv^2
where m is the mass of the ball, and v is the velocity. Potential energy is related to height by
PE = mgh
where g is the acceleration of gravity (we can assume we’re at the surface of the Earth, where g = 9.8 m/s^2), and h is height.
The law of conservation of energy says that the total energy is always the same. There are two states we’re concerned with:
State 1 has the ball moving at 4.70 m/s, at a height of 0 m.
State 2 has the ball moving at unknown velocity v2, at a height of .760 m.
To apply the law of conservation of energy, we simply calculate and equate the total energy in each state.
KE1 + PE1 = KE2 + PE2
1/2 * m * (4.70 m/s)^2 + m * 9.8 m/s^2 * 0 m = 1/2 * mv2^2 + m*9.8 m/s^2 * 0.760 m
You can cancel the mass of the ball, and you’ll find v2 = 2.68 m/s.
Hope that helps! If you want to make the problem a little more interesting, read on…
The solution above neglects one other kind of energy – rotational energy. As the ball slows down, its rotation slows down too, which means its rotational energy is changing. But in order to do anything with rotational energy, you need to know the diameter of the ball. Wikipedia says it’s 8.5 in, or 0.216 m (http://en.wikipedia.org/wiki/Bowling_ball). That means the radius, R, is 0.108 m.
The rotational kinetic energy is given by
KR = 1/2 * I * w,
(see http://hyperphysics.phy-astr.gsu.edu/Hbase/rke.html) where I is the mass moment of inertia of the ball, and w is its angular velocity (in a unit like rad/sec). If you assume the ball is a solid sphere, then the moment of inertia is
I = 2/5 * mR^2
(see http://hyperphysics.phy-astr.gsu.edu/Hbase/mi.html#mi).
To convert the velocity into angular velocity, first lets assume that the ball is rolling without slipping. Then every time it moves a linear distance equal to its circumference, it has rotated 2pi radians. The circumference is given by 2pi*R, so the linear distance can be related to the angular rotation by
distance/rotation = 2pi*R/(2pi)
distance/rotation = R
We can get velocities by dividing the top and bottom of the fraction by time:
(distance/time)/(rotation/time) = R
v/w = R
w = v/R
So, we can plug that into the rotation energy to get
KR = 1/2 * (2/5 * mR^2) * v/R
KR = 1/5 * mvR
Okay, now let’s use conservation of energy again, accounting for rotational energy.
KE1 + PE1 + KR1 = KE2 + PE2 + KR2
1/2 * m * (4.70 m/s)^2 + m*(9.8 m/s^2)*(0 m) + 1/5 * m * (4.70 m/s) * (0.108 m) =
1/2 * m * v2^2 + m*(9.8 m/s^2)*(0.760 m) + 1/5 * m * v2 * (0.108 m)
Once again, the mass cancels, and when we solve for v2, we get v2 = 2.70 m/s. So there’s not much error (only 0.02 m/s) in neglecting the rotation!
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