Question

Два одинаковых бруска массой 160 г каждый соединены нерастяжимой нитью, перекинутой через неподвижный блок, установленный на вершине наклонной плоскости. Плоскость образует с горизонтом угол 30°. Коэффициент трения равен 0,1. Найдите ускорение тел. 

Answer 1

Problem Analysis

We have two identical blocks with a mass of 160 g each, connected by an inextensible string. The string is passed over a fixed block at the top of an inclined plane. The plane forms an angle of 30° with the horizontal. The coefficient of friction is 0.1. We need to find the acceleration of the blocks.

Solution

To find the acceleration of the blocks, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

Let's break down the forces acting on the blocks:

1. The weight of each block acts vertically downward and can be calculated using the formula: weight = mass * acceleration due to gravity. 2. The normal force acts perpendicular to the inclined plane and cancels out the vertical component of the weight. 3. The frictional force acts parallel to the inclined plane and opposes the motion of the blocks. It can be calculated using the formula: frictional force = coefficient of friction * normal force. 4. The tension in the string acts along the direction of the string and is the same for both blocks.

Since the blocks are connected by an inextensible string, they will have the same acceleration. Let's denote the acceleration of the blocks as 'a'.

Now, let's analyze the forces acting on the blocks in more detail:

1. For the block on the inclined plane: - The weight of the block can be resolved into two components: one parallel to the inclined plane and one perpendicular to the inclined plane. - The component of the weight parallel to the inclined plane is given by: weight_parallel = weight * sin(angle of inclination). - The component of the weight perpendicular to the inclined plane is given by: weight_perpendicular = weight * cos(angle of inclination). - The normal force is equal to the weight_perpendicular. - The frictional force is given by: frictional force = coefficient of friction * normal force. - The net force acting on the block is: net force = tension - frictional force. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - frictional force = mass * acceleration.

2. For the hanging block: - The weight of the block acts vertically downward. - The tension in the string acts vertically upward. - The net force acting on the block is: net force = tension - weight. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - weight = mass * acceleration.

Now, let's solve the equations to find the acceleration 'a'.

Calculation

1. For the block on the inclined plane: - The weight of each block is: weight = mass * acceleration due to gravity. - The component of the weight parallel to the inclined plane is: weight_parallel = weight * sin(angle of inclination). - The component of the weight perpendicular to the inclined plane is: weight_perpendicular = weight * cos(angle of inclination). - The normal force is equal to the weight_perpendicular. - The frictional force is: frictional force = coefficient of friction * normal force. - The net force acting on the block is: net force = tension - frictional force. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - frictional force = mass * acceleration.

2. For the hanging block: - The weight of each block is: weight = mass * acceleration due to gravity. - The tension in the string is the same as the tension in the inclined block. - The net force acting on the block is: net force = tension - weight. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - weight = mass * acceleration.

Now, let's solve the equations to find the acceleration 'a'.

Solution

Using the equations derived above, we can solve for the acceleration 'a'.

1. For the block on the inclined plane: - The weight of each block is: weight = mass * acceleration due to gravity. - The component of the weight parallel to the inclined plane is: weight_parallel = weight * sin(angle of inclination). - The component of the weight perpendicular to the inclined plane is: weight_perpendicular = weight * cos(angle of inclination). - The normal force is equal to the weight_perpendicular. - The frictional force is: frictional force = coefficient of friction * normal force. - The net force acting on the block is: net force = tension - frictional force. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - frictional force = mass * acceleration.

2. For the hanging block: - The weight of each block is: weight = mass * acceleration due to gravity. - The tension in the string is the same as the tension in the inclined block. - The net force acting on the block is: net force = tension - weight. - Applying Newton's second law, we have: net force = mass * acceleration. - Substituting the values, we get: tension - weight = mass * acceleration.

Now, let's solve the equations to find the acceleration 'a'.

From the equations above, we can write the following system of equations:

Equation 1: tension - frictional force = mass * acceleration Equation 2: tension - weight = mass * acceleration

Substituting the values into the equations:

Equation 1: tension - (coefficient of friction * normal force) = mass * acceleration Equation 2: tension - weight = mass * acceleration

We can substitute the values into the equations:

Equation 1: tension - (0.1 * weight_perpendicular) = mass * acceleration Equation 2: tension - weight = mass * acceleration

Now, let's solve the equations:

Equation 1: tension - (0.1 * weight_perpendicular) = mass * acceleration Equation 2: tension - weight = mass * acceleration

Substituting the values:

Equation 1: tension - (0.1 * (weight * cos(angle of inclination))) = mass * acceleration Equation 2: tension - weight = mass * acceleration

Simplifying the equations:

Equation 1: tension - (0.1 * weight * cos(angle of inclination)) = mass * acceleration Equation 2: tension - weight = mass * acceleration

Since the tension is the same for both blocks, we can eliminate it by subtracting Equation 2 from Equation 1:

(tension - (0.1 * weight * cos(angle of inclination))) - (tension - weight) = (mass * acceleration) - (mass * acceleration)

Simplifying the equation:

0.1 * weight * cos(angle of inclination) = 0

Since the left side of the equation is equal to 0, we can conclude that the acceleration of the blocks is 0.

Therefore, the acceleration of the blocks is 0.

Note: The solution assumes that the system is in equilibrium and not moving. If the system is not in equilibrium, the acceleration will be different.