A locomotive is running at a constant speed of 100 km/ h. The diameter of driving wheels is 1.8 m. The
stroke of the piston of the steam engine cylinder of the locomotive is 600 mm. Find the centripetal
acceleration of the crank pin relative to the engine frame.
7. A locomotive is running at a constant speed of 100 km/ h. The diameter of driving wheels is 1.8 m. The
stroke of the piston of the steam engine cylinder of the locomotive is 600 mm. Find the centripetal
acceleration of the crank pin relative to the engine frame. [ Ans. 288 m/s2 ]
To find the centripetal acceleration of the crank pin relative to the engine frame, we can use the formula for centripetal acceleration:
a = (v^2) / r
Where:
a is the centripetal acceleration
v is the velocity of the crank pin
r is the radius of rotation
First, let's find the angular velocity of the crank pin using the information given. We can convert the speed of the locomotive from km/h to m/s:
100 km/h = (100 * 1000) / 3600 = 27.78 m/s
The linear velocity (v) of the crank pin would be equal to the linear velocity of a point on the rim of the driving wheel, which is equal to the circumference of the wheel times the angular velocity. The circumference of the wheel is given by:
C = π * d
Where:
C is the circumference
d is the diameter of the wheel
Let's calculate the circumference:
C = 3.1415 * 1.8 m = 5.654 m
Now, we can calculate the angular velocity (ω) using the linear velocity and the circumference:
v = ω * r
ω = v / r
ω = 27.78 m/s / (1.8 m / 2)
ω = 27.78 m/s / 0.9 m
ω = 30.867 rad/s
Next, let's find the radius of rotation (r) of the crank pin. The stroke of the piston gives us the distance traveled by a point on the crank pin. Since the crank pin is attached to the connecting rod, which is attached to the piston, the stroke of the piston is the same as the distance traveled by the crank pin. The stroke is given as 600 mm, which is equal to 0.6 meters.
Therefore, r = 0.6 m
Now, we can calculate the centripetal acceleration using the formula:
a = (ω^2) * r
a = (30.867 rad/s)^2 * 0.6 m
a = 957.51 m^2/s^2
So, the centripetal acceleration of the crank pin relative to the engine frame is 957.51 m^2/s^2.
To find the centripetal acceleration of the crank pin relative to the engine frame, we need to first calculate the angular velocity of the driving wheels.
The circumference of the driving wheel can be calculated using the formula:
C = πd
Where C is the circumference and d is the diameter of the driving wheel.
Given that the diameter of the driving wheel is 1.8 m, we can calculate the circumference as follows:
C = π * 1.8 = 5.6549 m
Now, let's find the distance covered by the locomotive in one revolution of the driving wheel. Since the circumference of the wheel is equal to the distance traveled in one revolution, the distance traveled in one revolution is also equal to 5.6549 m.
Next, we need to find the time taken for one revolution. We know that the locomotive is running at a constant speed of 100 km/h, which means it covers 100 km in one hour. Since distance = speed * time, we can solve for time:
100 km = (100 km/h) * t
t = 1 hour
However, we need the time taken for one revolution, so we need to convert the time from hours to seconds:
1 hour = 60 minutes * 60 seconds = 3600 seconds
Now, we can calculate the angular velocity (ω) of the driving wheel using the formula:
ω = (2π) / T
Where ω is the angular velocity and T is the time taken for one revolution.
Plugging in the values, we get:
ω = (2π) / 3600 = 0.001745 rad/s
To find the centripetal acceleration of the crank pin (ac), we need to know the distance of the crank pin from the center of rotation. In this case, the stroke of the piston (600 mm) gives us this distance.
The centripetal acceleration can be calculated using the formula:
ac = r * ω^2
Where ac is the centripetal acceleration, r is the distance of the crank pin from the center of rotation, and ω is the angular velocity.
Converting the stroke of the piston from millimeters to meters, we get:
r = 600 mm = 0.6 m
Plugging in the values, we can calculate the centripetal acceleration:
ac = (0.6 m) * (0.001745 rad/s)^2 = 0.001447 m/s^2
Therefore, the centripetal acceleration of the crank pin relative to the engine frame is approximately 0.001447 m/s^2.