The motion of a piston in an automobile engine is nearly simple harmonic. If the 1-kg piston travels back and forth over a distance of 12.0 cm, what is its maximum speed when the engine is running at 3000 rpm?
3000 rpm is a period of 1/50 second
So, with an amplitude of 6, the function is
y = 6 sin(100π t)
so, the speed is 600π cos(100π t), which has a maximum value of 600π cm/s
To solve this problem, we need to find the maximum speed of the piston when the engine is running at 3000 rpm.
Let's break it down into steps:
Step 1: Convert the engine speed from rpm to radians per second.
To do this, we need to multiply the engine speed by 2π (since there are 2π radians in one revolution) and divide by 60 (to convert from minutes to seconds).
Engine speed in radians per second = (3000 rpm) * (2π rad/rev) / (60 s/min)
Step 2: Calculate the angular frequency (ω) of the piston motion.
The angular frequency is equal to the engine speed in radians per second.
ω = Engine speed in radians per second
Step 3: Calculate the amplitude (A) of the piston's motion.
The amplitude A is given as the distance the piston travels back and forth, which is 12.0 cm.
Step 4: Calculate the maximum speed (v_max) of the piston.
The maximum speed of the piston is equal to the product of the angular frequency (ω) and the amplitude (A).
v_max = ω * A
Let's calculate the result:
Step 1: Convert the engine speed from rpm to radians per second.
Engine speed in radians per second = (3000 rpm) * (2π rad/rev) / (60 s/min)
Engine speed in radians per second = 100π rad/s (approximately)
Step 2: Calculate the angular frequency (ω) of the piston motion.
ω = Engine speed in radians per second = 100π rad/s (approximately)
Step 3: Calculate the amplitude (A) of the piston's motion.
The distance the piston travels back and forth is given as 12.0 cm.
Amplitude (A) = 12.0 cm = 0.12 m
Step 4: Calculate the maximum speed (v_max) of the piston.
v_max = ω * A = (100π rad/s) * (0.12 m)
v_max = 12π m/s (approximately)
Therefore, the maximum speed of the piston when the engine is running at 3000 rpm is approximately 12π m/s.
To find the maximum speed of the piston, we need to use the formula for the velocity of an object in simple harmonic motion (SHM). The formula is given as:
v = ω * A
Where:
- v is the velocity of the object (in m/s).
- ω is the angular frequency of the motion (in rad/s).
- A is the amplitude or maximum displacement of the object (in m).
First, let's calculate the angular frequency (ω) using the given information. The angular frequency is defined as the number of complete cycles or revolutions per unit time. In this case, the engine is running at 3000 revolutions per minute (rpm).
To convert from rpm to rad/s, we can use the following conversion factors:
1 revolution = 2π radians
1 minute = 60 seconds
So, the angular frequency (ω) can be calculated as follows:
ω = (2π * rpm) / 60
Plugging in the given value:
ω = (2π * 3000) / 60
Next, we can calculate the angular frequency (ω) using a calculator:
ω ≈ 314.16 rad/s
Now that we have the angular frequency (ω), we can calculate the maximum speed (v). Given that the piston travels back and forth over a distance of 12.0 cm, the amplitude or maximum displacement (A) is 0.12 m.
Plugging the values into the formula for velocity, we get:
v = ω * A
v = (314.16 rad/s) * (0.12 m)
v ≈ 37.7 m/s
Therefore, the maximum speed of the piston when the engine is running at 3000 rpm is approximately 37.7 m/s.