A screw jack with a lever arm of 40cm and a pitch 2.0cm is used to raise a heavy load. Find the velocity ratio of the screw jack
The velocity ratio (VR) of a screw jack can be calculated using the formula:
VR = (distance moved by effort) / (distance moved by load)
In this case, the effort is applied through the lever arm, which moves a distance equal to the circumference of the screw. The circumference of a screw can be calculated as:
C = πd
where d is the diameter of the screw. Since the pitch of the screw (P) is given as 2.0cm, we can express the diameter in terms of the pitch as:
d = P/π
Substituting the value of P = 2.0cm, we get:
d = 2.0/π ≈ 0.6366cm
Therefore, the circumference of the screw is:
C = πd ≈ 1.998cm
When the lever arm moves through one complete rotation (360°), the screw moves a distance equal to its pitch (P) along its axis. The distance moved by the effort (lever arm) is equal to the circumference of the circle described by its end, which can be calculated as:
C1 = 2πr
where r is the length of the lever arm (40cm). Substituting the value of r, we get:
C1 = 2π(40) ≈ 251.33cm
Therefore, the distance moved by the effort is:
distance moved by effort = C1 = 251.33cm
The load is raised by the screw a distance equal to its pitch (P) for each complete turn of the screw. Therefore, the distance moved by the load (lifting height) is equal to the product of the pitch (P) and the number of turns of the screw (n). The number of turns required to raise the load to a certain height depends on the thread pitch and the total weight of the load.
Let's assume that the load is raised by one full turn of the screw. Then the distance moved by the load is:
distance moved by load = P = 2.0cm
Finally, the velocity ratio is:
VR = (distance moved by effort) / (distance moved by load)
VR = 251.33cm / 2.0cm
VR ≈ 125.67
Therefore, the velocity ratio of the screw jack is approximately 125.67. This means that for each centimeter the lever arm is moved, the load will be raised by only 0.008cm (1/125.67 cm). However, the screw jack can provide a significant mechanical advantage when lifting heavy loads, since the effort required to turn the screw is spread out over a large distance.
To find the velocity ratio of the screw jack, we need to divide the distance moved by the effort applied by the distance moved by the load.
The distance moved by the effort can be found using the formula:
Distance moved by the effort = Circumference of the screw = π × diameter
The pitch of the screw is given as 2.0 cm, and the diameter can be calculated as follows:
Diameter = 2 × pitch = 2 × 2.0 cm = 4.0 cm
Now, we can calculate the circumference of the screw:
Circumference = π × diameter = 3.14 × 4.0 cm ≈ 12.56 cm
Therefore, the distance moved by the effort is 12.56 cm.
To find the distance moved by the load, we can use the formula:
Distance moved by the load = Pitch × Number of turns
Since the pitch is given as 2.0 cm, we need to determine the number of turns. This information is not provided in the question, so we cannot calculate the velocity ratio without this information.