To tackle this problem, we need to analyze the forces acting on the block in two different frames of reference: the non-inertial frame of the accelerating truck and the inertial frame of the ground. Let's break this down step by step.
Free Body Diagram (FBD) of the Block with Respect to the Truck
In the non-inertial frame of the truck, we need to consider the effects of the truck's acceleration. The block experiences a pseudo force due to the truck's acceleration. Here’s how the forces act on the block:
- Weight (W): The gravitational force acting downwards, which is equal to \( mg \) (where \( m \) is the mass of the block and \( g \) is the acceleration due to gravity).
- Normal Force (N): The force exerted by the truck surface on the block, acting perpendicular to the surface.
- Pseudo Force (F_p): This force acts horizontally in the opposite direction of the truck's acceleration, calculated as \( F_p = ma \) (where \( a \) is the acceleration of the truck).
The FBD will show the weight acting downwards, the normal force acting upwards, and the pseudo force acting horizontally backward (opposite to the direction of the truck's acceleration).
Free Body Diagram (FBD) of the Block with Respect to the Ground
In the inertial frame of the ground, the block experiences the following forces:
- Weight (W): Same as before, acting downwards.
- Normal Force (N): Acting upwards, equal to the weight of the block.
- Frictional Force (F_f): This force acts in the direction of the truck's acceleration, opposing the sliding of the block. It can be calculated using \( F_f = \mu N \), where \( \mu \) is the coefficient of friction.
In this frame, the block is subject to the truck's acceleration and the frictional force that tries to keep it from sliding.
Calculating the Acceleration of the Block
Now, let’s calculate the acceleration of the block in both frames of reference.
Acceleration with Respect to the Truck Frame
In the truck's frame, the block will experience a backward pseudo force due to the truck's acceleration. The net force acting on the block can be expressed as:
Net Force (F_net) = Pseudo Force - Frictional Force
Using the values provided:
- Mass of the block, \( m = 10 \, \text{kg} \)
- Acceleration of the truck, \( a = 4 \, \text{m/s}^2 \)
- Coefficient of friction, \( \mu = 0.2 \)
- Normal force, \( N = mg = 10 \times 9.81 = 98.1 \, \text{N} \)
- Frictional force, \( F_f = \mu N = 0.2 \times 98.1 = 19.62 \, \text{N} \)
Now, the pseudo force acting on the block is:
Pseudo Force = ma = 10 \times 4 = 40 \, \text{N}
Setting up the equation:
F_net = 40 - 19.62 = 20.38 \, \text{N}
Using Newton's second law, \( F = ma \), we can find the acceleration of the block with respect to the truck:
Acceleration (a_block) = F_net / m = 20.38 / 10 = 2.038 \, \text{m/s}^2
Acceleration with Respect to the Ground Frame
In the ground frame, the block is also subjected to the truck's acceleration. However, since it is sliding, we need to consider the frictional force acting in the direction of the truck's acceleration. The effective acceleration of the block can be calculated as:
Acceleration of the block (a_block_ground) = Acceleration of the truck - Acceleration due to friction
Since the frictional force is not enough to keep the block stationary relative to the truck, we can say:
Frictional acceleration = F_f / m = 19.62 / 10 = 1.962 \, \text{m/s}^2
Thus, the acceleration of the block with respect to the ground is:
a_block_ground = 4 - 1.962 = 2.038 \, \text{m/s}^2
Summary of Results
In summary:
- The acceleration of the block with respect to the truck frame is approximately 2.038 m/s².
- The acceleration of the block with respect to the ground frame is also approximately 2.038 m/s².
This analysis illustrates how the forces interact in different frames of reference and how friction plays a crucial role in determining the motion of the block on the accelerating truck.
