Solid A of mass 50 kg resting on a smooth 45º incline is maintained in equilibrium by means of the light string passing over the pulley and holding at its other end a solid B of mass m.
Make the inventory of the forces acting on A.
Determine graphically the magnitude of the tension in the string and the reaction of the incline (scale 1 cm =100cm) deduce the value of m.
To make an inventory of the forces acting on solid A, we need to consider all the forces that are acting on it.
1. Weight of solid A: This force acts vertically downwards and is equal to the mass of A (50 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). So, the weight of A is (50 kg) * (9.8 m/s^2) = 490 N.
2. Normal force: This force is perpendicular to the incline and counteracts the component of gravity acting along the incline. The normal force is equal in magnitude but opposite in direction to the component of the weight of A acting along the incline, which can be calculated using trigonometry. The component of the weight of A acting along the incline can be found using the equation: weight of A * sin(angle of incline). So, the normal force is 490 N * sin(45º) = 346 N.
3. Tension in the string: The tension in the string is the force exerted by the string to hold solid B. Since the system is in equilibrium, the tension in the string must balance the weight of solid B.
4. Reaction of the incline: This force acts perpendicular to the incline and counteracts the component of the weight of A acting perpendicular to the incline. The reaction force can be calculated using trigonometry. The component of the weight of A acting perpendicular to the incline is given by: weight of A * cos(angle of incline).
To determine the values of tension in the string and the reaction of the incline, we can create a free body diagram of solid A and use graphical methods.
Using a scale of 1 cm = 100 N, draw a vector representing the weight of A (490 N) acting vertically downwards from the center of solid A. Then, draw a vector representing the reaction of the incline (346 N) acting perpendicular to the incline, starting from the point of contact between A and the incline.
Next, draw a vector representing the tension in the string starting from the same point of contact and extending in the direction of solid B. Connect the end of this vector to the tail end of the vector representing the weight of A. The magnitude of this resulting vector will represent the tension in the string.
To find the value of m (mass of solid B), measure the length of the resultant vector for the tension in the string using the scale (1 cm = 100 N) and convert it to Newtons. Divide this value by the acceleration due to gravity (9.8 m/s^2) to get the mass, m.
Using these graphical methods, you can determine the magnitude of the tension in the string and deduce the value of m.