Question

(a) At what rate must you lift a 3.6-kg container of milk (1 gallon) if the power output of your arm is to be 22 W? (b) How long does it take to lift the milk container through a distance of 1.0 m at this rate?

Short answer

(a) Lift at 0.627 m/s. (b) It takes 1.59 seconds.

Step-by-step solution

Understand the Relationship between Power, Force, and Velocity

The power output of an object is given by the formula: \[ P = F \cdot v \] where \( P \) is the power in watts, \( F \) is the force in newtons, and \( v \) is the velocity in meters per second. Here, we're asked to find the velocity (rate of lifting) at which a 3.6 kg container is lifted with a power of 22 W.

Calculate the Force Required to Lift the Container

The force required to lift the container is equal to its weight, which can be calculated using: \[ F = m \cdot g \] where \( m = 3.6 \) kg is the mass and \( g = 9.8 \) m/s² is the acceleration due to gravity. Thus, \( F = 3.6 \times 9.8 \).

Solve for the Velocity

Using the power formula \( P = F \cdot v \), we solve for velocity \( v \): \[ v = \frac{P}{F} \] Substituting the known values gives: \[ v = \frac{22}{3.6 \times 9.8} \].

Calculate the Time to Lift 1 Meter

Use the formula for velocity: \[ v = \frac{d}{t} \] where \( d = 1.0 \) m is the distance to be lifted and \( t \) is the time in seconds. Solving for \( t \) gives: \[ t = \frac{d}{v} \]. Substitute the value of \( v \) from Step 3.

Key concepts

Force

Force is a fundamental concept in physics. It refers to any interaction that changes the motion of an object. In terms of lifting an object, such as our 3.6 kg container of milk, force is essential for overcoming gravity.
Let's explore this with the formula for force when lifting:

• Force: Given by the equation \[ F = m \cdot g \], where
• \( m \) is mass (3.6 kg for the milk container)
• \( g \) is acceleration due to gravity (\( 9.8 \text{ m/s}^2 \) on Earth).

The force calculated helps us understand how much effort is needed to lift the milk against gravity. In our case, this equates to supporting the weight of the container.

Velocity

Velocity is a vector quantity that denotes the speed and direction of an object. It's crucial when determining how quickly an object needs to be moved or lifted. For our milk container, we need to find the rate at which it must be elevated when exerting a specific power.
Using the formula for power, \[ P = F \cdot v \], we solve for velocity as:

• \( v = \frac{P}{F} \)
• Power (\( P \)) is given (22 W in this scenario),
• and Force (\( F \)) was calculated previously.

Calculating velocity gives us how rapidly the container must be lifted to use the provided power efficiently.

Weight

Weight is the force exerted by gravity on an object. It's frequently confused with mass, but they are different. Weight is mass affected by gravity's pull.
Let’s break down how we calculate the weight of the container:

• Weight equals the force exerted by gravity: \[ F = m \cdot g \]
• In our example, the 3.6 kg milk container’s weight is computed by multiplying its mass by the gravitational constant (\( 9.8 \text{ m/s}^2 \)

Weight gives us a clear picture of the force needed to counteract gravity's pull and lift the object.

Acceleration due to gravity

Acceleration due to gravity is a constant that affects all objects on Earth. This constant value, approximately \( 9.8 \text{ m/s}^2 \), ensures that objects fall at the same rate when only gravitational forces act on them.
It's important for understanding how gravity impacts force and weight:

• It's the factor used alongside mass to determine the force required to lift objects (\( F = m \cdot g \)
• Acts uniformly on all masses, giving them weight based on their mass.

This constant enables us to calculate the necessary force and energy expenditure in lifting exercises.