To solve the given problems, we'll use the principles of work-energy and conservation of energy. Let's address each question one by one:
(1) What is the work done by Chris on the box when the speed of the box reaches 2.1 m/s?
The work done by Chris on the box is equal to the change in the box's kinetic energy. Since the box starts from rest, the initial kinetic energy is zero. The final kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) * mass * velocity^2
Plugging in the values:
Mass of the box (m) = 53 kg
Final velocity (v) = 2.1 m/s
Kinetic energy = (1/2) * 53 kg * (2.1 m/s)^2
Calculate the value of the kinetic energy, which represents the work done by Chris on the box.
(2) What is the speed of the box when it reaches the bottom of the slope (Point B in the diagram)?
To determine the speed at the bottom of the slope, we'll use the principle of conservation of energy. The total mechanical energy of the box is conserved as it moves from the top to the bottom of the slope.
The initial potential energy at the top of the slope is converted into kinetic energy at the bottom of the slope, neglecting any energy losses due to friction.
Potential energy at the top = m * g * h1
Where:
Mass of the box (m) = 53 kg
Acceleration due to gravity (g) = 10 m/s^2
Height difference between floors (h1) = 3.2 m
Calculate the initial potential energy.
The final kinetic energy at the bottom is given by:
Kinetic energy at the bottom = (1/2) * m * v^2
Where:
Mass of the box (m) = 53 kg
Velocity at the bottom (v) = ?
Equating the initial potential energy to the final kinetic energy, solve for v to find the speed of the box at the bottom of the slope.
(3) To what speed does the box slow down when it reaches the bottom of the ramp to the pickup truck?
Since the ramp connecting the first floor to the bed of the pickup truck is frictionless, there is no external force doing work on the box. Thus, the mechanical energy of the box is conserved as it moves from the bottom of the slope to the bottom of the ramp.
Using the same principle of conservation of energy, equate the final kinetic energy at the bottom of the slope to the initial potential energy at the bottom of the ramp.
Potential energy at the bottom of the ramp = m * g * h2
Where:
Mass of the box (m) = 53 kg
Acceleration due to gravity (g) = 10 m/s^2
Height difference between the first floor and the truck bed (h2) = 0.90 m
Calculate the potential energy at the bottom of the ramp.
Equating the potential energy at the bottom of the ramp to the final kinetic energy, solve for the speed of the box at the bottom of the ramp.
(4) What is the speed of the box when it reaches the bed of the pickup truck?
Since the ramp connecting the first floor to the truck bed is frictionless, there is no external force doing work on the box. The mechanical energy of the box is conserved as it moves from the bottom of the ramp to the truck bed.
Using the same principle of conservation of energy, equate the final potential energy at the bottom of the ramp to the final kinetic energy at the truck bed.
Potential energy at the truck bed = m * g * h