A particle of mass m is driven by a machine that delivers a constant power k watt.

If the particle starts from rest the force on the particle at time t is

• A. √(mk) t⁻¹/²
• B. √(2mk) t⁻¹/²
• C. 1/2 √(mk) t⁻¹/²
• D. √(mk/2) t⁻¹/²

To determine the force acting on a particle of mass m driven by a machine delivering constant power k watts, we can use the relationship between power, force, and velocity. Power (P) is defined as the product of force (F) and velocity (v):

Power and Force Relationship

The equation can be expressed as:

P = F * v

Velocity as a Function of Time

Since the particle starts from rest, we can find the velocity as a function of time. The power delivered is constant, so we can express it in terms of mass and acceleration:

Using the equation for power:

P = m * a * v

We can relate acceleration (a) to force (F) using Newton's second law:

F = m * a

Finding the Force

From the power equation, we can isolate force:

F = P / v

Now, we need to express velocity in terms of time. Since the particle starts from rest and is subjected to constant power, we can derive:

v = (2 * P / m)^(1/2) * t^(1/2)

Substituting Back to Find Force

Substituting this expression for velocity back into the force equation gives:

F = P / [(2 * P / m)^(1/2) * t^(1/2)]

After simplifying, we find:

F = √(mk) * t^(-1/2)

Final Answer

The correct answer is:

A. √(mk) t⁻¹/²