2.3 Newton’s second law A frictional force can also occur when a force acts on a body but the body does not move, in which case we speak of static friction. There is no formula for the static frictional force, but there is one for the largest possible frictional force that can develop between two bodies. This is fmax = μsN, where μs is the coefficient of static friction and N is the magnitude of the normal reaction force between the bodies. Note that this formula gives the maximum possible static frictional force, not the frictional force in general.
Worked Example 2.3 A block of weight 3.0 N rests on an inclined plane. The angle of the incline is 30 °. The coefficient of dynamic friction between the block and the plane is 0.58. a Calculate the static frictional force between the plane and the block.
N
b The angle of the incline is slowly increased. When the angle barely exceeds 42°, the body begins to slide down the plane. Determine i the static coefficient of friction between the body and the plane, and ii the net force on the block as it slides down the plane. Figure 2.10 shows the forces on the block.
fs
θ θ
mg
Figure 2.10
a Analysing the weight into components along and normal to the plane, we find that, at equilibrium: mg sin θ = fs N = mg cos θ Hence, fs = mg sin θ = 3.0 × sin30 = 1.5N. (Note that the expression fs = μs N gives the maximum possible static frictional force. The actual static frictional force is not always as large as this.) b i Because the largest angle at which equilibrium occurs is 42°, the frictional force in this case must be the largest possible and so is given by fmax = µs N . So we have that: mg sin 42 = µs N N = mg cos 42 This leads to mg sin 42 = µs mg cos 42 and so µs = tan 42 = 0.90. ii The frictional force is now the dynamic frictional force since we have sliding. The angle of the incline is just barely larger than 42°. Since N = mg cos 42 = 3.0 × cos 42 = 2.23N, the frictional force is fd = µdN = 0.58 × 2.23 = 1.29 ≈ 1.3N. The net force is then mg sin 42 − fd = 3.0 × sin 42 − 1.29 = 0.717 ≈ 0.72N .
2.3 Newton’s second law Newton’s second law states that the net force on a body of constant mass is equal to the product of the mass and its acceleration: Fnet = ma. Note that F and a are vector quantities and are in the same direction.
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