91Ó°ÊÓ

Gravitational force plays a vital role in various physics problems, especially when dealing with bodies moving on inclines.

Gravitational force is defined by the equation: \( F_{grav} = m \times g \times \sin(\theta) \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (usually \( 9.8 \, m/s^2 \) on Earth's surface), and \( \theta \) is the angle of the incline. In this case, for a mass of \( 950 \, kg \) on a \( 30^\circ \) slope, the component of gravitational force acting along the slope is \( 4637.75 \, N \) as determined by the given formula.

This force must be countered by the winch motor to move the car up the slope at a constant speed, or to start its movement from rest.

Power is a measure of the rate at which work is performed or energy is transferred. In physics, it is often calculated as the product of force times velocity, according to the formula \( P = F \times v \).

When the ore car is moving with uniform velocity, it means that the accelerative forces have balanced out and the only force the motor needs to account for is the gravitational force along the slope. At a constant speed of \( 2.20 \, m/s \) and a force of \( 4637.75 \, N \), the necessary power output of the motor is \( 10203.05 \, W \) or roughly \( 10.2 \, kW \). This power is required to maintain the car’s steady speed up the incline.

Uniform acceleration refers to the constant rate of change of velocity that an object experiences. The formula to find acceleration is \( a = \frac{v - u}{t} \), where \( v \) is the final velocity, \( u \) is the initial velocity, and \( t \) is the time taken to reach the final velocity from rest.

For the ore car, accelerating uniformly from rest to \( 2.20 \, m/s \) in \( 12 \, s \) results in an acceleration of \( 0.183 \, m/s^2 \). The constancy of this acceleration simplifies the calculations involved in determining the forces and power required during this period of the car’s journey.

The work-energy principle is a key concept in physics that states that the work done on an object is equal to the change in its kinetic energy. Work is calculated by the formula \( W = F \times d \) where \( F \) is the force applied and \( d \) is the distance over which the force is applied.

In the context of our problem, the total work done by the winch motor when the car moves \( 1250 \, m \) is \( 6014500 \, J \) or \( 6014.5 \, kJ \). This signifies the total energy supplied by the motor to move the car along the entire track. This principle is foundational in understanding how energy is harnessed and converted to do work in mechanical systems.