A bowling ball of mass 7.3 kg and radius 9.6 cm rolls without slipping down a lane at 3.4 m/s. Calculate its total kinetic energy. I tried this problem with the equation 3/4mv^2 but I did not get the correct answer, could I please get help with step-by-step explanation of how to get the correct equation to use.
The total KE is 7/10 M V^2. The rotational part is 1/5 M V^2, and the translational part is 1/2 MV^2.
Get the rotational KE from the moment of inertia (2/5)MR^2, KErot = (1/2) I w^2, and V = R w
Oh, the poor bowling ball trying to get its kinetic energy right! Let me help you out with a step-by-step explanation.
To calculate the total kinetic energy of the rolling bowling ball, we need to consider both its translational kinetic energy and its rotational kinetic energy.
First, let's calculate the translational kinetic energy. The equation for translational kinetic energy is:
KE_trans = 1/2 * m * v^2
Where:
- KE_trans is the translational kinetic energy
- m is the mass of the bowling ball
- v is the velocity of the bowling ball
Plugging in the values you provided, we get:
KE_trans = 1/2 * 7.3 kg * (3.4 m/s)^2
Now, let's calculate the rotational kinetic energy. The equation for rotational kinetic energy is:
KE_rot = 1/2 * I * ω^2
Where:
- KE_rot is the rotational kinetic energy
- I is the moment of inertia of the bowling ball (which depends on its shape and mass distribution)
- ω is the angular velocity
For a solid spherical object like a bowling ball, the moment of inertia can be calculated as:
I = 2/5 * m * r^2
Where:
- r is the radius of the bowling ball
Plugging in the values you provided, we get:
I = 2/5 * 7.3 kg * (0.096 m)^2
Now, we need to find the angular velocity. For a rolling object without slipping, the angular velocity is related to the linear velocity by the equation:
ω = v / r
Plugging in the values you provided, we get:
ω = 3.4 m/s / 0.096 m
Now we have everything we need to calculate the rotational kinetic energy:
KE_rot = 1/2 * (2/5 * 7.3 kg * (0.096 m)^2) * ((3.4 m/s / 0.096 m)^2)
Finally, we can find the total kinetic energy by summing up the translational and rotational kinetic energies:
Total KE = KE_trans + KE_rot
That's it! Just plug everything into the equations, do the calculations, and you'll have the correct answer. Remember, practice makes perfect, so don't slip up on your next attempt!
To calculate the total kinetic energy of the bowling ball, you need to consider both its translational kinetic energy and rotational kinetic energy.
1. Translational kinetic energy:
The formula to calculate translational kinetic energy is:
Kinetic energy (KE) = 1/2 * mass * velocity^2
In this case, the mass of the bowling ball is given as 7.3 kg, and the velocity is 3.4 m/s. Plugging these values into the formula, we get:
KE_translational = 1/2 * 7.3 kg * (3.4 m/s)^2
Simplifying the equation:
KE_translational = 1/2 * 7.3 kg * 11.56 m^2/s^2
KE_translational = 39.908 J (rounded to three decimal places)
So, the translational kinetic energy of the bowling ball is approximately 39.908 Joules.
2. Rotational kinetic energy:
The formula to calculate rotational kinetic energy is:
KE_rotational = 1/2 * moment of inertia * angular velocity^2
However, the moment of inertia depends on the shape of the object. For a uniform solid sphere, the moment of inertia (I_s) is given by:
I_s = 2/5 * mass * radius^2
In this case, the mass of the bowling ball is still 7.3 kg, and the radius is given as 9.6 cm. But the radius should be in meters, so we convert it:
Radius = 9.6 cm = 0.096 m
Now, we can calculate the moment of inertia:
I_s = 2/5 * 7.3 kg * (0.096 m)^2
Simplifying the equation:
I_s = 2/5 * 7.3 kg * 0.009216 m^2
I_s = 0.05997312 kg*m^2 (rounded to eight decimal places)
Next, we need to find the angular velocity (ω). For an object rolling without slipping, the linear velocity (v) is related to the angular velocity by the equation:
v = ω * radius
Rearranging the equation to solve for angular velocity:
ω = v / radius
Plugging in the values:
ω = 3.4 m/s / 0.096 m
ω = 35.4167 rad/s (rounded to four decimal places)
Now we can calculate the rotational kinetic energy:
KE_rotational = 1/2 * 0.05997312 kg*m^2 * (35.4167 rad/s)^2
Simplifying the equation:
KE_rotational = 1/2 * 0.05997312 kg*m^2 * 1257.9474 rad^2/s^2
KE_rotational = 37.394 J (rounded to three decimal places)
So, the rotational kinetic energy of the bowling ball is approximately 37.394 Joules.
3. Total kinetic energy:
To find the total kinetic energy, you simply need to add the translational and rotational kinetic energies together:
Total KE = KE_translational + KE_rotational
Total KE = 39.908 J + 37.394 J
Total KE = 77.302 J (rounded to three decimal places)
Therefore, the total kinetic energy of the bowling ball is approximately 77.302 Joules.
To calculate the total kinetic energy of the bowling ball, you need to consider both its translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
The translational kinetic energy is given by the equation KE_trans = 1/2 * m * v^2, where m is the mass of the bowling ball and v is its velocity.
However, since the bowling ball is rolling without slipping, it also has rotational kinetic energy. The rotational kinetic energy is given by the equation KE_rot = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
To determine the moment of inertia of a solid sphere (like a bowling ball) rolling without slipping, you can use the equation I = 2/5 * m * r^2, where r is the radius of the bowling ball.
Now, let's calculate the total kinetic energy step by step.
Step 1: Calculate the translational kinetic energy (KE_trans).
KE_trans = 1/2 * m * v^2
= 1/2 * 7.3 kg * (3.4 m/s)^2
= 1/2 * 7.3 kg * 11.56 m^2/s^2
≈ 43.538 Joules
Step 2: Calculate the rotational kinetic energy (KE_rot).
I = 2/5 * m * r^2
= 2/5 * 7.3 kg * (0.096 m)^2
= 2/5 * 7.3 kg * 0.009216 m^2
≈ 0.07313824 kg * m^2
Since the bowling ball is rolling without slipping, its translational velocity (v_t) is related to its angular velocity (ω) by the equation v_t = ω * r.
Given v_t = 3.4 m/s and r = 0.096 m, we can solve for ω:
3.4 m/s = ω * 0.096 m
ω = 3.4 m/s / 0.096 m
ω ≈ 35.4167 rad/s
Now, calculate the rotational kinetic energy using the equation KE_rot = 1/2 * I * ω^2.
KE_rot = 1/2 * 0.07313824 kg * m^2 * (35.4167 rad/s)^2
≈ 43.53797 Joules
Step 3: Calculate the total kinetic energy (KE_total).
KE_total = KE_trans + KE_rot
≈ 43.538 Joules + 43.53797 Joules
≈ 87.075 Joules
Therefore, the total kinetic energy of the bowling ball is approximately 87.075 Joules.