9.1.3: Wedges

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Key Questions

• Why is the normal force always perpendicular to the contact surface while the friction force always lies parallel to it?

• Can you demonstrate graphically that components of friction and normal forces are related to the right-triangle trigonometry terms sine and cosine?

A wedge is a tapered object which converts a small input force into a large output force using the principle of an inclined plane. Wedges are used to separate, split or cut objects, lift weights, or fix objects in place. The mechanical advantage of a wedge is determined by the angle of its taper; narrow tapers have a larger mechanical advantage.

Wedges are used in two primary ways:

Low friction wedges are a simple machines which allows users to create large output forces to move objects using comparatively small input forces. In the log splitter in Figure 9.3.1.(a), hydraulic ram pushes a log into a stationary wedge. The normal force pushes the two halves of the log apart while the friction force \(F_f\) is opposes the pushing force \(P\).

High-friction (self-locking) wedges control the location of objects or hold them in place. Examples include doorstop wedges and carpentry wedges. The sailor in Figure 9.3.1.(b) is hammering two wooden wedges towards each other to create large compressive forces to secure shoring timbers during a damage control operation.

                                        

(a) A low friction wedge is used to                                        (b) High friction wedges are used split logs to secure shoring timbers.

Figure 9.3.1. Wedges in use.

 

Luckily the analysis of low- and high-friction wedges are identical and they are quite similar to the multi-force body equilibrium problems we saw in Chapter 5 and Chapter 6. The main difference is the inclusion of friction from all non-smooth contact surfaces. The directions of both the normal and friction forces on the free-body diagrams are defined below.

Normal forces act between bodies act perpendicular to the contacting surfaces. All normal forces on a free-body diagram should be pointing towards the body because wedges are always subjected to compression.

Friction forces are between bodies which act parallel or tangential to the contacting surfaces and are created by the microscopic or large scale roughness of the surfaces. All friction forces on a free-body diagram should be drawn pointing in the direction which resists relative motion at the point of contact.

The key added challenge of solving wedge problems is that the angled faces of wedges usually need to be resolved into components in the x and y unless a different coordinate system is used.

One of the critical steps in solving block or wedge problems is to determine which force is engaging the friction of the system. Start by drawing the friction forces on the body where this force acts. As you pass the friction and normal forces to adjacent free-body diagrams, you must always show them as equal and opposite, action-reaction pairs. This is illustrated in the following example.

Example \(\PageIndex{1}\)

Find the minimum force P required to start raising the 10 lb block. Assume that the wedge is massless.

Solution

      

This figure demonstrates the free-body diagrams to find the minimum \(P_1\) to raise the block. We would assume that all the friction forces are pushed to impending motion, thus you can use \(F=\mu_{\mathrm{s}} N\) to relate the friction and normal forces at all contact surfaces. A detail of \(N_2\) and \(F_2\) has also been provided so that you can see how the angle \(\theta\) is incorporated into the \(x\) and \(y\) components.

Example 9.3.3.

Using the same system as the previous problem

If the problem in Example \(\PageIndex{1}\) was changed to "Given a coefficient of static friction of \(\mu_{\mathrm{s}}=0.6\) find the minimum force \(P_2\) to keep the wedge from slipping out under the \(10 \mathrm{lb}\) block", the free-body diagrams would need to change in the following ways:

• all friction force directions would change as the impending motion of both the wedge and \(10 \mathrm{lb}\) block would change direction and
• the direction of \(P\) may have to change if the wedge has sufficient friction to stay static when \(P=0\).

Note that for all values of \(P\) between \(P_1\) and \(P_2\) the system would be static, and the friction forces would be static-but-not-impending.