Question

Брусок массой 5 кг движется под горку с наклоном 40° с коэффициентом трения 0,65 при наличие внешней силы 20H, направленной вдоль наклонной поверхности в горку. Найдите величину и направление ускорения. Нарисуйте чертеж с изображением силы тяжести, веса, силы реакции опоры, скатывающей силы, внешней силы и силы трения.

Answer 1

M*a=F+m*g*sina-k*m*g*cosa
a=F+m*g*(sina-k*cosa)/m=20+5*10*(0,64-0,65*0,77)/5=5,4 м/с2

Answer 2

Problem Analysis

We are given the following information: - Mass of the block: 5 kg - Inclination angle of the slope: 40° - Coefficient of friction: 0.65 - External force: 20 N directed along the inclined surface towards the top of the slope

We need to find: - Magnitude and direction of acceleration - Drawing with the forces acting on the block, including gravity, weight, reaction force, rolling force, external force, and friction force.

Solution

To solve this problem, we'll break it down into the following steps:

1. Calculate the weight of the block. 2. Calculate the component of the weight parallel to the slope. 3. Calculate the normal force. 4. Calculate the rolling force. 5. Calculate the net force. 6. Calculate the friction force. 7. Calculate the acceleration. 8. Determine the direction of acceleration. 9. Draw a diagram showing the forces acting on the block.

Let's go through each step in detail.

Step 1: Calculate the weight of the block

The weight of an object is given by the formula: weight = mass × acceleration due to gravity. Given that the mass of the block is 5 kg, and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight as follows: weight = 5 kg × 9.8 m/s².

Step 2: Calculate the component of the weight parallel to the slope

The component of the weight parallel to the slope can be calculated using the formula: component = weight × sin(angle). In this case, the angle is 40°. So, the component of the weight parallel to the slope is: component = weight × sin(40°).

Step 3: Calculate the normal force

The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the component of the weight perpendicular to the slope. The normal force can be calculated using the formula: normal force = weight × cos(angle). In this case, the angle is 40°. So, the normal force is: normal force = weight × cos(40°).

Step 4: Calculate the rolling force

The rolling force is the force that opposes the motion of the block. It can be calculated using the formula: rolling force = coefficient of friction × normal force. In this case, the coefficient of friction is 0.65, and the normal force is calculated in the previous step.

Step 5: Calculate the net force

The net force is the vector sum of all the forces acting on the block. In this case, the net force is the sum of the external force and the rolling force. The net force can be calculated using the formula: net force = external force - rolling force. In this case, the external force is given as 20 N, and the rolling force is calculated in the previous step.

Step 6: Calculate the friction force

The friction force is the force that opposes the motion of the block. It can be calculated using the formula: friction force = coefficient of friction × normal force. In this case, the coefficient of friction is 0.65, and the normal force is calculated in step 3.

Step 7: Calculate the acceleration

The acceleration of the block can be calculated using Newton's second law of motion: acceleration = net force / mass. In this case, the net force is calculated in step 5, and the mass of the block is given as 5 kg.

Step 8: Determine the direction of acceleration

The direction of acceleration can be determined by comparing the net force and the friction force. If the net force is greater than the friction force, the block will accelerate in the direction of the net force. If the net force is less than the friction force, the block will decelerate or move in the opposite direction. In this case, compare the magnitudes of the net force and the friction force to determine the direction of acceleration.

Step 9: Draw a diagram showing the forces acting on the block

Draw a diagram showing the forces acting on the block, including gravity, weight, reaction force, rolling force, external force, and friction force.

Now, let's calculate the values and draw the diagram.

Step 1: Calculate the weight of the block

The weight of the block is given by the formula: weight = mass × acceleration due to gravity. Given that the mass of the block is 5 kg, and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight as follows: weight = 5 kg × 9.8 m/s² = 49 N.

Step 2: Calculate the component of the weight parallel to the slope

The component of the weight parallel to the slope can be calculated using the formula: component = weight × sin(angle). In this case, the angle is 40°. So, the component of the weight parallel to the slope is: component = 49 N × sin(40°) ≈ 31.5 N.

Step 3: Calculate the normal force

The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the component of the weight perpendicular to the slope. The normal force can be calculated using the formula: normal force = weight × cos(angle). In this case, the angle is 40°. So, the normal force is: normal force = 49 N × cos(40°) ≈ 37.5 N.

Step 4: Calculate the rolling force

The rolling force is the force that opposes the motion of the block. It can be calculated using the formula: rolling force = coefficient of friction × normal force. In this case, the coefficient of friction is 0.65, and the normal force is calculated in the previous step. rolling force = 0.65 × 37.5 N ≈ 24.4 N.

Step 5: Calculate the net force

The net force is the vector sum of all the forces acting on the block. In this case, the net force is the sum of the external force and the rolling force. The net force can be calculated using the formula: net force = external force - rolling force. In this case, the external force is given as 20 N, and the rolling force is calculated in the previous step. net force = 20 N - 24.4 N ≈ -4.4 N.

Step 6: Calculate the friction force

The friction force is the force that opposes the motion of the block. It can be calculated using the formula: friction force = coefficient of friction × normal force. In this case, the coefficient of friction is 0.65, and the normal force is calculated in step 3. friction force = 0.65 × 37.5 N ≈ 24.4 N.

Step 7: Calculate the acceleration

The acceleration of the block can be calculated using Newton's second law of motion: acceleration = net force / mass. In this case, the net force is calculated in step 5, and the mass of the block is given as 5 kg. acceleration = -4.4 N / 5 kg ≈ -0.88 m/s².

Step 8: Determine the direction of acceleration

The direction of acceleration can be determined by comparing the net force and the friction force. If the net force is greater than the friction force, the block will accelerate in the direction of the net force. If the net force is less than the friction force, the block will decelerate or move in the opposite direction. In this case, the magnitude of the net force (4.4 N) is less than the magnitude of the friction force (24.4 N). Therefore, the block will decelerate or move in the opposite direction of the net force.

Step 9: Draw a diagram showing the forces acting on the block

The diagram below shows the forces acting on the block: ``` |<---- 31.5 N ---->| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |