a bowling ball has a mass of 8kg. if it rolls down the alley, without slipping, at 7m/s, calculate a) the linear kinetic energy, b) the rotaional kinetic energy, c) the total kinetic engergy
The maximum legal mass of a bowlng ball is actually 7.27 kg, but who's counting?
a) (1/2)MV ^2, of course
b) (1/2) I w^2
Use (2/5)M R^2 for the moment of inertia, I, and V/R for the angular velocity w.
You will get an expression in terms of M and V. R will cancel out.
c) Add the results of a and b
You should get (7/10)M*V^2
To calculate the linear kinetic energy of the bowling ball, we use the formula:
a) Linear kinetic energy (K_linear) = (1/2) * mass * velocity^2
Given:
Mass (m) = 8 kg
Velocity (v) = 7 m/s
Substituting the values into the formula:
K_linear = (1/2) * 8 kg * (7 m/s)^2
K_linear = 28 J (Joules)
b) To calculate the rotational kinetic energy of the bowling ball, we use the formula:
Rotational kinetic energy (K_rotational) = (1/2) * moment of inertia * angular velocity^2
Since the ball is rolling without slipping, the moment of inertia (I) for a solid sphere can be calculated as:
I = (2/5) * mass * radius^2
Given:
Mass (m) = 8 kg
For a standard bowling ball, the radius is 0.11 m (approximately).
Substituting the values and calculating the moment of inertia:
I = (2/5) * 8 kg * (0.11 m)^2
I ≈ 0.195 kg·m²
We need to find the angular velocity (ω), which is equal to the linear velocity divided by the radius:
ω = v / r
Given:
v = 7 m/s
r ≈ 0.11 m
Substituting the values:
ω = 7 m/s / 0.11 m
ω ≈ 63.64 rad/s
Now, let's calculate the rotational kinetic energy:
K_rotational = (1/2) * 0.195 kg·m² * (63.64 rad/s)^2
K_rotational ≈ 395.11 J (Joules)
c) The total kinetic energy (K_total) is just the sum of the linear and rotational kinetic energies:
K_total = K_linear + K_rotational
K_total = 28 J + 395.11 J
K_total ≈ 423.11 J (Joules)
Therefore, the linear kinetic energy is 28 J, the rotational kinetic energy is approximately 395.11 J, and the total kinetic energy is approximately 423.11 J.
To calculate the linear kinetic energy (KE), rotational kinetic energy (KE_rot), and total kinetic energy (KE_total) of a rolling bowling ball, you need to use the equations and formulas for each type of kinetic energy. Here's how to calculate each one:
a) Linear kinetic energy (KE):
The formula to calculate linear kinetic energy is:
KE = 0.5 * m * v^2
where KE is the linear kinetic energy, m is the mass of the object, and v is the velocity.
In this case, the mass of the bowling ball is given as 8 kg and the velocity as 7 m/s. Plugging these values into the formula, we get:
KE = 0.5 * 8 kg * (7 m/s)^2
Simplifying further:
KE = 0.5 * 8 kg * 49 m^2/s^2
KE = 196 kg.m^2/s^2
So, the linear kinetic energy of the bowling ball is 196 kg.m^2/s^2.
b) Rotational kinetic energy (KE_rot):
The formula to calculate rotational kinetic energy is:
KE_rot = 0.5 * I * ω^2
where KE_rot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.
For a solid sphere like a bowling ball, the moment of inertia is given by:
I = (2/5) * m * r^2
where r is the radius of the bowling ball.
In this case, the mass of the bowling ball is given as 8 kg, and a typical bowling ball has a uniform density, which means the radius can be calculated using the mass and density. Let's assume the density of the bowling ball is 1,000 kg/m^3.
Density = mass / volume
1,000 kg/m^3 = 8 kg / (4/3 * π * r^3)
Solving for r:
r^3 = (4/3 * π * 8 kg) / 1,000 kg/m^3
r^3 = 10.24 m^3 / 1,000 kg/m^3
r^3 ≈ 0.01024 m^3
r ≈ (0.01024)^(1/3) m
r ≈ 0.215 m
Plugging the radius into the moment of inertia formula:
I = (2/5) * 8 kg * (0.215 m)^2
Simplifying further:
I ≈ 0.276 kg.m^2
Since the ball is rolling without slipping, the angular velocity (ω) is related to the linear velocity (v) by:
v = ω * r
Rearranging the equation, we can solve for ω:
ω = v / r
ω = 7 m/s / 0.215 m
ω ≈ 32.56 rad/s
Plugging the values of I and ω into the formula for rotational kinetic energy, we get:
KE_rot = 0.5 * 0.276 kg.m^2 * (32.56 rad/s)^2
Simplifying further:
KE_rot ≈ 145.4 kg.m^2/s^2
So, the rotational kinetic energy of the bowling ball is approximately 145.4 kg.m^2/s^2.
c) Total kinetic energy (KE_total):
The total kinetic energy is the sum of the linear kinetic energy and the rotational kinetic energy. So, to find the total kinetic energy, we add the values obtained in parts (a) and (b):
KE_total = KE + KE_rot
KE_total = 196 kg.m^2/s^2 + 145.4 kg.m^2/s^2
Calculating the sum:
KE_total ≈ 341.4 kg.m^2/s^2
Therefore, the total kinetic energy of the rolling bowling ball is approximately 341.4 kg.m^2/s^2.