A power screw has a lead angle of 3° and a friction angle of 12°. An engineer redesigns the thread to increase the lead angle to 15°. What changes about the screw's behavior?
AThe screw becomes more self-locking because a larger lead angle traps more friction force
BThe screw loses self-locking — the load can now back-drive the screw because the lead angle exceeds the friction angle
CThe screw lifts loads faster but retains self-locking because the friction coefficient is unchanged
DNothing changes — self-locking depends only on the friction coefficient, not the thread geometry
Self-locking requires that the lead angle be less than the friction angle (φ = arctan μ). At 3° < 12°, the screw is self-locking: friction holds the load without external input. At 15° > 12°, the geometry places the reaction force outside the friction cone, so the load can back-drive the thread — the screw is no longer self-locking. This design tradeoff is critical: a car jack must be self-locking (you need to leave it holding the car), while a lead screw on an adjustable instrument may need to be back-drivable.
Question 2 Multiple Choice
A band brake wraps around a drum with wrap angle β = π radians and friction coefficient μ = 0.3. The slack side tension is 50 N. What is the approximate tight side tension? (Use e^(0.3π) ≈ 2.57)
A50 N — friction makes no difference at the wrap angle used
B65 N — tension scales linearly with wrap angle: T = T_slack × (1 + μβ)
D500 N — the tight side is always ten times the slack side in standard brakes
The capstan equation T_tight/T_slack = e^(μβ) is exponential, not linear. Each infinitesimal element of the belt adds a friction contribution proportional to the current tension, so the contributions compound multiplicatively around the wrap. With μ = 0.3 and β = π: T_tight = 50 × e^(0.942) ≈ 50 × 2.57 ≈ 128 N. Option B represents the common mistake of assuming linear scaling. The exponential nature is what makes a few wraps of rope around a post capable of holding a very large load with modest input force.
Question 3 True / False
A wedge is self-locking when its wedge angle is greater than the friction angle.
TTrue
FFalse
Answer: False
Self-locking occurs when the wedge angle is *less than* the friction angle. When the wedge angle < φ (friction angle), the geometry forces the reaction force inside the friction cone, so friction is always sufficient to prevent sliding regardless of the applied load. When the wedge angle > φ, the reaction force falls outside the friction cone and the wedge slides under load. The self-locking criterion — angle < friction angle — is the same for wedges, power screws, and any device where inclined surfaces interact.
Question 4 True / False
A power screw thread is geometrically equivalent to a wedge wrapped around a cylinder, so the self-locking criterion (lead angle vs. friction angle) applies to both.
TTrue
FFalse
Answer: True
The analogy is direct: unrolling a screw thread produces an inclined plane (wedge) whose slope is the lead angle. The same self-locking condition applies: if the helix (lead) angle is less than the friction angle, the load cannot unscrew the thread — the screw self-locks. This is why a car jack keeps a vehicle raised without holding the handle, and why highly-efficient ball-screw actuators (with very low friction angles) are deliberately not self-locking and require a brake to hold position.
Question 5 Short Answer
Explain the self-locking criterion for a wedge or power screw. Why does a car jack not require you to hold the handle to keep the car raised?
Think about your answer, then reveal below.
Model answer: Self-locking occurs when the lead (or wedge) angle is less than the friction angle φ = arctan(μ). When this condition holds, any load trying to back-drive the device creates a reaction force that falls within the friction cone — friction can always generate enough force to resist motion, so the device stays put without external input. A car jack is designed with a thread whose lead angle is below the friction angle, so the weight of the car cannot unscrew the jack; the geometry locks it in place.
The friction angle is the maximum angle at which the reaction force can be directed away from the normal while friction still holds. When the device geometry forces the reaction force to stay within this cone, no sliding is possible regardless of load magnitude — this is self-locking. Engineers exploit this in jacks, clamps, and turnbuckles while deliberately choosing lead angles above the friction angle when back-drivability is needed (adjustable instruments, certain actuators).