a person pushes a 8 kg cart up a 13 degree incline. Find the magnitude of the horizontal force needed to give the cart an acceleration of 1.55 m/s^2.
forceupplane=mass*acceleration
forceupplane=forcehorizontal*cos13
forcehorizontal=
=mass*acceleation/cos13=8*1.55/cos13deg
To find the magnitude of the horizontal force needed to give the cart an acceleration of 1.55 m/s^2, we can break the force into its horizontal and vertical components.
First, we need to determine the weight of the cart. The weight can be calculated as the product of the mass and the acceleration due to gravity. The acceleration due to gravity on Earth is approximately 9.8 m/s^2. Therefore, the weight of the cart is given by:
weight = mass * acceleration due to gravity
= 8 kg * 9.8 m/s^2
= 78.4 N
Next, let's study the forces acting on the cart when it is on the inclined plane. There are two main forces at play: the weight of the cart acting vertically downward, and the force applied parallel to the incline that we need to find in order to achieve the given acceleration.
Since the cart is on an incline, we can resolve the weight into two components: one acting perpendicular to the incline, and the other acting parallel to the incline. The component parallel to the incline is the force that we need to find.
Now, we need to determine the component of the weight parallel to the incline, which can be calculated using trigonometry. The component of the weight parallel to the incline is given by:
component parallel to the incline = weight * sin(angle of inclination)
= 78.4 N * sin(13 degrees)
= 17.03 N
This component of the weight parallel to the incline represents the force acting in the opposite direction to the applied force.
To find the magnitude of the horizontal force needed, we need to overcome this component of the weight and provide an additional force to achieve the desired acceleration. Therefore, the magnitude of the horizontal force needed is:
magnitude of horizontal force = component parallel to the incline + force for desired acceleration
= 17.03 N + (mass * acceleration)
= 17.03 N + (8 kg * 1.55 m/s^2)
= 17.03 N + 12.4 N
= 29.43 N
Hence, the magnitude of the horizontal force needed to give the cart an acceleration of 1.55 m/s^2 is approximately 29.43 N.