1.5.2 Turning effect of forces (3)
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1.
A plank of length 4 m rests on a wall, with one end on the ground. The plank is supported by a rope tied to the ground 1.5 m from the base of the wall. The plank makes an angle of 60° with the ground. Calculate the tension in the rope.
Solution:
The plank acts as a lever. We need to consider the forces acting on the plank and the pivot point (where it rests on the wall). The forces are: the weight of the plank (acting downwards at the midpoint of the plank), the tension in the rope (acting upwards at the point where it's tied to the ground), and the reaction force from the wall (acting upwards at the point where the plank rests against the wall).
First, find the weight of the plank. We need to assume a weight for the plank. Let's assume the plank has a weight of 200 N acting downwards at the midpoint (2m from either end).
The distance from the wall to the point where the rope is tied is 1.5m. The distance from the wall to the midpoint of the plank is 2m.
The moment due to the weight of the plank is: 200 N x 2 m = 400 Nm.
The moment due to the tension in the rope is: T x 1.5 m.
For equilibrium: 400 Nm = T x 1.5 m
Therefore, T = 400 Nm / 1.5 m = 266.67 N
The tension in the rope is approximately 266.67 N.
2.
An object is resting on a horizontal surface. A vertical force of 20N is applied to the object at a point 0.5m from the edge. A horizontal force of 15N is also applied to the object at the same point. Assuming the object has a mass of 5kg and the surface is frictionless, explain why the object remains at rest. In your answer, clearly define the terms 'resultant force' and 'resultant moment'.
Answer: An object remains at rest when there is no resultant force acting upon it and no resultant moment about any point. In this scenario, the vertical force of 20N and the horizontal force of 15N act on the object. The resultant force is the vector sum of all forces. The vertical component of the resultant force is 20N. The horizontal component of the resultant force is 15N. Since these forces are not equal and opposite, the resultant force is not zero. Therefore, the object would accelerate. However, the question states that the object remains at rest. This implies that there must be some other force acting on the object that counteracts the resultant force. The most likely explanation is that there is an upward reaction force from the surface of the object that is equal in magnitude and opposite in direction to the applied forces. This reaction force would balance the resultant force, resulting in a resultant force of zero.
A resultant force is the vector sum of all forces acting on an object. It determines the acceleration of the object. A resultant moment is the vector sum of all moments about a chosen point. It determines the rotation of the object.
3.
A seesaw is balanced by a boy of mass 30kg sitting 1.5m from the fulcrum and a girl of mass 20kg sitting 2.5m from the fulcrum. State, with reference to the principle of equilibrium, why the seesaw is balanced. Show your calculations to support your answer.
Answer: The seesaw is balanced because the resultant moment about the fulcrum is zero. For the seesaw to be in equilibrium, the clockwise moment must equal the anticlockwise moment.
The moment is calculated as Force x Distance. In this case, the force is the weight of the person (mass x gravitational acceleration, g = 9.8 m/s2).
Boy's moment: Moment = 30kg x 9.8 m/s2 x 1.5m = 441 kNm (clockwise)
Girl's moment: Moment = 20kg x 9.8 m/s2 x 2.5m = 490 kNm (anticlockwise)
The seesaw is balanced when the clockwise moment equals the anticlockwise moment. However, in this case, the boy's moment (441 kNm) is less than the girl's moment (490 kNm). This indicates an error in the question or the provided data. Assuming the question is correct, the seesaw would not be balanced. However, if the girl's position was adjusted so that the moment was equal to the boy's, the seesaw would be in equilibrium. The principle of equilibrium states that an object is in equilibrium if and only if the resultant force is zero and the resultant moment is zero.