Chapter 02 Equilibrium and Muscles PDF

by  CHAPTER 2 Equilibrium and Muscles Force & Torque Static & Equilibrium Equilibrium in the Human Body Muscles and Tendons Levers Back muscles Elbow Force & Torque Force and torque are both vectors and fundamental concepts in physics. They...

by  CHAPTER 2 Equilibrium and Muscles Force & Torque Static & Equilibrium Equilibrium in the Human Body Muscles and Tendons Levers Back muscles Elbow Force & Torque Force and torque are both vectors and fundamental concepts in physics. They describe different ways objects interact and move. Force A push or pull exerted on an object that change its state of motion. Can cause an object to start moving, speed up, slow down, change direction, or deform. Measured in Newton (N). Represented by an arrow indicating the direction and magnitude of the push or pull and generally is calculated from; mass × acceleration. Force could be the weight W of something (mass × acceleration of gravity = mg) and pulls downwards. If this object is placed on a table (for example) there will be a normal reaction force from the table on the object upwards. Force & Torque Torque The twisting or turning effect created by a force applied at a distance from an object's Large arm, axis of rotation. large torque Imagine using a wrench to tighten a nut, the force you apply creates torque, causing the nut to rotate. Measured in Newton. Meters (N⋅m). Small arm, small torque Calculated by multiplying the perpendicular force by the distance from the point of rotation to the point where the force is applied. Force & Torque The magnitude of torque τ equals the applied force F times the length of arm r that is perpendicular to the applied force. τ = 𝐹𝑟 sin 𝜃 where 𝜃 is the angle between the direction of the force and the direction of 𝑟. Torque is zero if the force is passing through the pivot point, in other words, parallel to the arm (i.e., 𝜃=0). Torque is maximum if the force is perpendicular to the arm (i.e., 𝜃=90). The direction of the torque is obtained by the right- hand rule as shown. Quick Quiz Calculate the torques (magnitudes and directions) in both figures. Answer: 𝜏 = 𝐹𝑟 sin 𝜃 𝜏 = 𝐹𝑟 sin 𝜃 −2 −2 = 17 × 25 × 10 × sin 37 = ⋯ 𝑁𝑚 = 2.5 × 15 × 10 × sin 90 = ⋯ 𝑁𝑚 Rotation is counter-clockwise, Rotation is counter-clockwise, and direction is out of the page and direction is out of the page Static and Equilibrium A body is said to be mechanically static (not dynamic) if the linear and angular velocities are zero. A body is said to be in mechanical equilibrium if the next two conditions are verified: The resultant external force must equal zero 𝑭 = 𝟎. The resultant external torque about any axis must equal zero 𝝉 = 𝟎 otherwise, the body will rotate. The special condition that is called stable equilibrium, means that the sum of torques and the sum of forces, tend to make the body to return to its stable positions. Center of Mass and Stability The center of mass for an object is the average position of all object parts, according to their masses. The center of mass is a point in an object or system where its mass can be considered to be concentrated. If the gravity is constant all over the object parts, the center of mass can be considered as the center of gravity. For the stable body, the reaction force Fr (upwards) cancels the force produced by the body weight Fw (downwards) as they work in the same line with opposite directions  Conditions for Stable Equilibrium When the body is unstable, i.e., rotates around a pivoting point, there are two conditions: If its center of mass is above its base, the reaction force at the pivoting point and force produced by the body weight try to restore the body to its original position. If the center of mass is outside the base, the produced torque tends to topple the body. A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net torque in a direction opposite to the direction of the displacement. A system is unstable if, when displaced from equilibrium, it experiences a net torque in the same direction as the displacement from equilibrium. Static, Stable Equilibrium and Unstable Static Stable Equilibrium Unstable This pencil is in the condition If the pencil is displaced slightly to If the pencil is displaced of equilibrium. The net force the side (counterclockwise), it is counterclockwise too far, the torque on the pencil is zero and the not static. Its weight produces a caused by its weight changes total torque about any pivot is clockwise torque that returns the direction to counterclockwise and zero. The pencil is static. pencil to its equilibrium position. causes the displacement to increase. Equilibrium in the Human Body The center of gravity of an erect person with arms at the side is at approximately 56% of the person’s height measured from the soles of the feet. Note: The center of mass changes its position if the shape of the body changes, even if, the mass is kept constant. Quiz Calculate the height of a person whose center of gravity is at 112 cm measured from the soles of the feet. Answer The center of gravity is at 56% of the person’s height then 100 𝑇ℎ𝑒 𝑝𝑒𝑟𝑠𝑜𝑛’𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 = 112 × = 200 𝑐𝑚 = 2 𝑚 56  Equilibrium in the Human Body When carrying an uneven load, the body tends to compensate by bending and extending the limbs to shift the center of gravity back over the feet. People who have lost an arm often have some problems (i.e., permanent distortion of the spine), because of the continuous compensatory bending of the torso. Therefore, the amputees should wear an artificial arm to restore balanced weight distribution. (a) A father carrying his son piggyback leans forward to position their overall cg above the base of support at his feet. (b) A student carrying a shoulder bag leans to the side to keep the overall cg over his feet. (c) Another student carrying a load of books in her arms leans backward for the same reason. External Force on Human Body Stability Problem Let us assume a person as shown in Fig. Calculate the force applied to the shoulder to topple a person standing at rigid attention (Comparing the torques about point A). Assuming that the mass of the person is 70 kg (the gravitational acceleration g= 9.8 m/s2) Answer The anti-clockwise torque Ta is produced by the applied force Fa by pivoting around point A is 𝑇𝑎 = 𝐹𝑎 × 𝑟𝑎 = 𝐹𝑎 × 1.5 The opposite restoring torque Tw due to the person’s weight W is 𝑇𝑊 = 𝐹𝑤 × 𝑟𝑤 = 𝑊 × 𝑟𝑤 = 𝑚𝑔 × 𝑟𝑤 = 70 × 9.8 × 0.1 = 68.6 N. m The person is on the verge of toppling when the magnitudes of these two torques are equal 𝑇𝑎 = 𝑇𝑊 𝐹𝑎 × 1.5 = 68.6 The force required to topple an erect person is larger than 𝐹𝑎 = 68.6/1.5 = 45.7 𝑁 External Force on Human Body Stability Example A person is standing at rigid attention with mass of 90 kg. Its height measured from his shoulder is 120 cm and his foot width is 10 cm. Calculate the magnitude of the applied external force 2 required to topple this person. Assume the gravitational acceleration to be 10 m/s. Answer The anti-clockwise torque Ta is 𝑇𝑎 = 𝐹𝑎 × 𝑟𝑎 = 𝐹𝑎 × 1.2 The opposite restoring torque Tw due to the person’s weight W is 0.1 + 0.9 + 0.1 𝑇𝑊 = 𝐹𝑤 × 𝑟𝑤 = 𝑊 × 𝑟𝑤 = 𝑚𝑔 × 𝑟𝑤 = 90 × 10 × = 495 N. m 2 The person is on the verge of toppling when 𝑇𝑎 = 𝑇𝑊 𝐹𝑎 × 1.2 = 495 The force required to topple an erect person is larger than 𝐹𝑎 = 495/1.2 = 412.5 𝑁 Note: Spreading the legs makes the base of support wider which increases the stability against a toppling force (i.e., increases the restoring torque produced by the weight Tw). Muscles and Tendons The muscles producing skeletal movements consist of thousands of parallel fibers. The force exerted by the muscle depends on the number of fibers contracting the muscle. The thousands of fibers in each muscle give a high variability of the force. The tendons attach the muscle to the bones. Most muscles end in a single tendon. Except for biceps and triceps, they end in two and three tendons, respectively, and each tendon is attached to a different bone. Levers Levers are used for lifting loads in an advantageous way and to transfer movement from one point to another. A lever is a rigid bar free to rotate about a fixed point called the fulcrum. The position of the fulcrum is fixed with respect to the bar. There’s always a reaction force acting on the fulcrum to overcome the forces acting on the lever so as to keep the fulcrum point stable. This force has no torque as its arm is zero. There are three classes of levers as shown in the figure. Levers If the lever is in equilibrium, then, the torque produced by the applied force F is equal to the torque produced by the load W. if the load and force arms are d1 and d2, respectively then 𝑭 𝒅𝟐 = 𝑾 𝒅𝟏 By defining the mechanical advantage of the lever as the ratio of the weight to the applied force, we get 𝑾 𝒅𝟐 𝑴= = 𝑭 𝒅𝟏 Levers The three different classes examples of levers as shown. Levers In a Class 1 lever, the fulcrum is located between the applied force and the load. So, with d1 much smaller than d2, a large mechanical advantage M can be obtained. In a Class 2 lever, the load is located between the applied force and the fulcrum. Here, d1 is always smaller than d2 so the mechanical advantage is M >1. In a Class 3 lever, the applied force is located between the load and the fulcrum. d1 is always larger than d2 so the mechanical advantage is M