By Bradley Parker,2014-07-08 10:44
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    Putting one’s foot down


    In this activity you will read an article called ‘Forces can be larger than expected’ and use

    your knowledge of vectors to understand and explain the meaning of various terms. You will show your understanding by drawing a force diagram and calculating forces using moments and vector addition. You will gain experience in naming forces and drawing them in the correct position and direction.

    Forces can be larger than expected

    Read the text that follows and then answer the questions.

    Free body diagrams

    Drawing a free body diagram is a technique for investigating parts of a system rather than the whole of the system. The part in question is treated as the ‘free body’ and all the forces acting on it are drawn across the boundary surface that separates it from the rest of the body.

    Consider the free body analysis of the foot of someone standing on tiptoe on one leg. The ‘free body’ is the foot and the boundary of the free body is through the junction of the foot bones with the leg bones and the Achilles tendon. This tendon is pulled by the calf muscle. When the muscles are tightened, they produce a pulling force or tension T. In this example, it

    is an upward force. The ground also pushes upwards on the toe, with a force usually known as the reaction force R. If the foot is in equilibrium then the leg bones must push downwards with a force F. This is shown above in the ‘free body force analysis’ diagram.

    If the person is standing still, their weight W must be equal to R. This can be shown with a

    free body analysis of the whole body.

    At equilibrium, F = T + R. The pull of the calf muscle adds to the reaction force so that the force in the leg bones is larger than the weight. If the person is jumping upwards then the reaction R must be greater than the person’s weight and the force in the leg bones is even larger.

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Frictional forces

    Frictional forces are necessary during walking and running to prevent the foot slipping.

    Heel-strike occurs as the heel first touches the ground.

    The force acting up on the leg is the resultant of the frictional force F

    and the reaction force R of the ground.

    Because of the geometry of the vector diagram, when the two forces F

    and R are added together the resultant force is larger than either.

    This means that the force acting up on the leg is larger than the simple reaction force acting upwards from the ground. On an average walking surface the largest that F can be before the

    foot starts to slip is 15% greater than R.


    1 The article mentions three reasons why the total force in the leg bones can be larger than the weight of a person. What are they?

    2 What is meant by ‘equilibrium’ (beginning of paragraph 4 of ‘Free body diagrams’) and

    what are the conditions for equilibrium?

    3 The diagram below shows the situation when a person carries a weight in one hand. C is the centre of gravity of the hand and forearm.

    (a) On the diagram, draw a free body force analysis for the hand and forearm. Your

    diagram should show four labelled forces. Show clearly the direction of the force on

    the forearm caused by the bone in the upper arm, the humerus.

    (b) Considering the elbow as a pivot, suggest why the force in the biceps muscle is larger

    than the total weight of the arm and the weight being carried.

     4 When a ballerina stands on tiptoe the reaction force R is 400N. The dimensions of her

    foot are shown in the free body force analysis diagram in the article.


    (a) the force T in the Achilles tendon

    (b) the force F in the leg bones.

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    5 (a) Draw a vector diagram showing how the force acting on the leg during heel-strike is

    related to the frictional force and the reaction force on the ground. (b) Explain why a vector diagram shows that ‘when the two forces F and R are added

    together the resultant force is larger than either’. (Hint: see second-to-last paragraph.)

    6 The leg of a man first makes an angle of 30? with the vertical during heel-strike. The reaction force upwards due to the ground is 1000N.

    You may assume that, at heel-strike, the resultant force acting up on the leg acts along the direction of the leg.

    (a) Calculate the frictional force between the heel and the ground.

    (b) Calculate the force acting up on the leg.

    Is the heel likely to slip when it hits at this angle? (Hint: use the last sentence in the


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     To check comprehension of concepts of forces and simple ideas of equilibrium.

     To draw a free body force diagram and use moments to calculate forces.

     To draw a vector diagram and use this to calculate forces.

    Teaching notes

    Students should have covered moments and addition of vectors before tackling this exercise. To draw a free body force diagram, they may need to be shown how to isolate part of the body and consider the forces acting on the various parts in isolation. When taking moments, students should be encouraged to state clearly the point about which they take moments. Some students will need help in deciding the point about which to take moments where the ‘pivot’ is not clear.


    7 The force in the muscle adds to the force due to the ground; when jumping upwards, extra force is needed for the acceleration; when landing, frictional force adds to the reaction force of the ground.

    8 Equilibrium is a stable condition, where the total force on the body and the total moment on the body are both zero.

    9 (a) The diagram should show: three forces on the forearm acting downwards (the weight

    of the arm acting at C, the weight of the object in the hand and the reaction force on

    the humerus, which acts downwards at the elbow); and one force acting upwards,

    along the biceps muscle.

    (c) The biceps muscle is closer to the elbow so, to create the same moment, the force in

    the biceps muscle must be larger than the total weight of the arm and the weight

    being carried. Considered another way, there is a reaction force downwards at the

    elbow which adds to the weight and must be balanced by the force in the biceps


    10 (a) The principle of moments states that if a body is in equilibrium, the sum of the

    clockwise moments equals the sum of the anticlockwise moments. Stating this AQA Physics A AS Level Extension Activity ? Nelson Thornes Ltd 2008 4

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    another way, the sum of the moments about any point should be zero. Taking

    moments about the junction of the leg and foot bones (that is, where F acts):

    400 15 = T 6 So, T = 1000N

    (d) Moments can be taken about where the tendon meets the foot bone or about the toe

    but it is easier just to recognise that the upward forces must balance the downward


     F = T + R = 1000 + 400 = 1400N

    11 (a) The vector diagram should comprise a simple right-angled triangle with F horizontal,

    R vertical and the force acting on the leg as the hypotenuse. The arrows on the two

    forces added together must be ‘head to tail’ as shown in the following diagram.

    Alternatively, a parallelogram of forces can be drawn.

    (e) The two forces are at right angles and so the resultant is the hypotenuse in a right-

    angled triangle. The hypotenuse is always larger than either of the sides of a triangle.

     12 (a) In the vector triangle, F = 1000 tan30? = 577N

    1000 (f) Resultant force acting up on the leg = = 1155N cos30;

    or Pythagoras’ theorem may be used.

     (g) 15% larger means 1.15 times larger so R + 15% = 1000 1.15 = 1150N. Since F = 580N, it is less than R + 15% so the heel is not likely to slip at this angle.

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