What minimum constant force f has to be applied in the horizontal direction to the mass M1 in order to shift mass M2?

To determine the minimum constant force \( f \) that needs to be applied to mass \( M_1 \) in order to shift mass \( M_2 \), we need to consider the forces acting on both masses and the friction between them. This scenario typically involves two masses connected in some way, often with one mass resting on top of the other or adjacent to it. Let's break this down step by step.

Understanding the Forces at Play

When we apply a force \( f \) to mass \( M_1 \), it will create a frictional force between \( M_1 \) and \( M_2 \). The frictional force is what keeps \( M_2 \) stationary. To shift \( M_2 \), the applied force must overcome this frictional force.

Frictional Force Calculation

The frictional force \( F_f \) can be calculated using the formula:

• Frictional Force: \( F_f = \mu \cdot N \)

Where:

• \( \mu \) is the coefficient of friction between the two masses.
• \( N \) is the normal force acting on \( M_2 \), which is typically equal to the weight of \( M_2 \) if it is resting on a horizontal surface.

Normal Force Consideration

The normal force \( N \) can be expressed as:

• Normal Force: \( N = M_2 \cdot g \)

Here, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). Thus, the frictional force becomes:

• Frictional Force: \( F_f = \mu \cdot (M_2 \cdot g) \)

Applying the Force

For mass \( M_2 \) to start moving, the applied force \( f \) must be greater than the frictional force:

• Condition for Movement: \( f > F_f \)

Substituting the expression for the frictional force, we have:

• Minimum Force Required: \( f > \mu \cdot (M_2 \cdot g) \)

Final Expression

Therefore, the minimum constant force \( f \) that must be applied to mass \( M_1 \) to shift mass \( M_2 \) can be expressed as:

• Minimum Force: \( f = \mu \cdot (M_2 \cdot g) \)

Example for Clarity

Let’s say we have:

• Mass \( M_2 = 5 \, \text{kg} \)
• Coefficient of friction \( \mu = 0.4 \)

Calculating the normal force:

• \( N = 5 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 49.05 \, \text{N} \)

Now, calculating the frictional force:

• \( F_f = 0.4 \cdot 49.05 \, \text{N} = 19.62 \, \text{N} \)

Thus, the minimum force \( f \) required to shift \( M_2 \) would be:

• \( f = 19.62 \, \text{N} \)

In summary, by understanding the relationship between the applied force, friction, and the normal force, we can effectively determine the minimum force needed to initiate movement in a system involving two masses. This approach is fundamental in physics, especially in mechanics, where forces and motion are analyzed.