A square threaded screw jack of mean diameter 25mm and pitch of 6mm is used to lift the load of 150 Newton .find the torque generated at the mean circumference if the coefficient of friction between the screw and nut is 0.02.
The torque generated can be calculated using the equation:
Torque = Frictional Force x Mean Circumference
First, let's calculate the frictional force.
Frictional Force = Coefficient of Friction x Normal Force
Normal Force = Load / (2 x π x (Mean Diameter / 2))
Normal Force = 150 N / (2 x π x (25 mm / 2))
Normal Force = 150 N / (2 x π x 12.5 mm)
Normal Force = 150 N / (78.54 mm)
Normal Force = 1.907 N
Frictional Force = 0.02 x 1.907 N
Frictional Force = 0.03814 N
Next, let's calculate the mean circumference.
Mean Circumference = π x Mean Diameter
Mean Circumference = π x 25 mm
Mean Circumference = 78.54 mm
Now we can calculate the torque.
Torque = 0.03814 N x 78.54 mm
Torque = 2.999 N·mm
The torque generated at the mean circumference is 2.999 N·mm.
To find the torque generated at the mean circumference, we need to calculate two components: the effort force and the Frictional force.
1. Calculation of Effort Force:
The effort force (F) can be calculated using the formula:
F = Load / Mechanical Advantage
The mechanical advantage (MA) of a square threaded screw jack is given by:
MA = π x Mean Diameter / Pitch
Substituting the given values:
Mean Diameter = 25 mm
Pitch = 6 mm
Load = 150 Newtons
MA = π x 25 / 6
MA ≈ 4.12
Therefore, the effort force (F) can be calculated as:
F = 150 / 4.12
F ≈ 36.41 N
2. Calculation of Frictional Force:
The frictional force (Fr) can be calculated using the formula:
Fr = Coefficient of Friction x Normal Force
The normal force (N) is the force acting perpendicular to the contact surface and can be determined by:
N = Load / Number of Threads
In this case, since it is not mentioned, we assume a single start thread.
Therefore, N = Load.
Substituting the given values:
Coefficient of Friction = 0.02
Load = 150 Newtons
Therefore, the frictional force (Fr) can be calculated as:
Fr = 0.02 x 150
Fr = 3 N
3. Calculation of Torque (T):
The torque (T) generated at the mean circumference can be calculated using the formula:
T = (F - Fr) x (Mean Diameter / 2)
Substituting the values we have calculated:
F = 36.41 N
Fr = 3 N
Mean Diameter = 25 mm
Therefore, the torque (T) can be calculated as:
T = (36.41 - 3) x (25 / 2)
T = 33.41 x 12.5
T ≈ 417.63 N·mm
Thus, the torque generated at the mean circumference is approximately 417.63 N·mm.
To find the torque generated at the mean circumference, we need to calculate the force required to overcome the friction between the screw and the nut.
The force required to overcome friction can be calculated using the formula:
Force = Coefficient of Friction * Normal Force
First, let's find the normal force exerted on the screw jack. The normal force is equal to the weight being lifted by the screw jack.
Given:
Load = 150 Newtons
Next, we need to calculate the pitch circumference of the screw. The pitch circumference is the distance covered by the screw in one complete rotation.
Given:
Mean diameter = 25mm
Pitch = 6mm
The formula for the pitch circumference is:
Pitch Circumference = π * Mean Diameter
Let's calculate the pitch circumference:
Pitch Circumference = π * 25mm
Pitch Circumference ≈ 3.1416 * 25mm
Pitch Circumference ≈ 78.54mm
Now, let's find the torque by multiplying the force required to overcome friction by the pitch circumference:
Torque = Force * Pitch Circumference
To calculate the force required to overcome friction, we use the formula:
Force = Coefficient of Friction * Normal Force
Now, let's substitute the given values into the formula:
Force = 0.02 * 150N
Force ≈ 3N
Finally, we can calculate the torque:
Torque = 3N * 78.54mm
Torque ≈ 235.62 Nmm
Therefore, the torque generated at the mean circumference is approximately 235.62 Nmm.