Problem 9 - Applied Physics

Question

How long would it take a \(95 \overline{0}-\mathrm{W}\) motor to raise a \(36 \overline{0}-\mathrm{kg}\) mass to a height of \(16.0 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

It takes approximately 59.4 seconds to raise the mass to the height.

Step 1: Understand the problem

We have a motor with a power of 950 Watts that needs to lift a mass of 360 kg to a height of 16.0 meters. We need to determine how much time this process will take.

Step 2: Calculate the work needed

Work required to lift the mass is given by the equation: Work = Force x Distance. The force required to lift the object is equal to its weight, which is the mass multiplied by the acceleration due to gravity. Hence, Work = \( 360 \times 9.8 \times 16 = 56448 \) Joules.

Step 3: Relate work to power and time

Power is defined as the work done per unit of time. The formula for power is \( \text{Power} = \frac{\text{Work}}{\text{Time}} \). Rearranging this formula to solve for time gives \( \text{Time} = \frac{\text{Work}}{\text{Power}} \).

Step 4: Calculate the time

Substitute the known values into the formula for time: \( \text{Time} = \frac{56448}{950} = 59.4 \) seconds.

Key Concepts

Work-Energy Principle

The Work-Energy Principle is a fundamental concept in physics that describes how work done on an object is related to its energy. When we talk about work, we mean the energy transferred to or from an object via force. In essence, work leads to changes in the energy of an object.

In our exercise, the motor performs work by lifting a 360 kg mass to a height of 16 meters. Work is calculated using the formula:

• Work = Force x Distance

The force in this case is equal to the weight of the mass, calculated by multiplying the mass with the acceleration due to gravity (9.8 m/s²). By calculating the product, we find that the work done here amounts to 56,448 Joules.

This principle illustrates the connection between work, energy, and force. When energy in the form of work is applied to lift the object against gravity, it increases the object's potential energy, which is the energy associated with its position.

Power in Physics

Power, from a physics standpoint, is the rate at which work is done or energy is transferred over time. It measures how quickly work can be performed. The formula to determine power is:

• Power = Work / Time

In the given exercise, a motor with power rating of 950 Watts helps in performing the task of moving the object.

Using the power formula, you can rearrange it to find time if you know the work done and the power applied. This gives us:

• Time = Work / Power

Power plays a vital role in determining how fast an operation can be completed, especially in mechanical applications like lifting, where the amount of energy used over a specific time period is critical. A motor with higher power can finish the same task more rapidly compared to one with lower power.

Time Calculation in Mechanics

Understanding how to calculate time in mechanics showcases the practical application of the relationship between work, power, and time. This concept helps us determine how long it will take a given force to perform work under certain conditions. In mechanics, time determination often involves knowing the power available and the work required.

For tasks involving lifting or moving objects, like in our example, you would use:

• Time = Work / Power

By substituting the calculated work (56,448 Joules) and the power of the motor (950 Watts), the time required to lift the mass to the specified height is 59.4 seconds.

Learning how to calculate time using energy principles is essential in various engineering and physics applications. It enables predictions about how long processes will take, which is vital for planning and efficiency.