A person pushes a 10.5-kg shopping cart at a constant velocity for a distance of 25.7 m on a flat horizontal surface. She pushes in a direction 30.9 ° below the horizontal. A 35.3-N frictional force opposes the motion of the cart. (a) What is the magnitude of the force that the shopper exerts? Determine the work done by (b) the pushing force, (c) the frictional force, and (d) the gravitational force.
To find the magnitude of the force that the shopper exerts, we need to decompose the force into horizontal and vertical components.
Given:
Mass of the shopping cart, m = 10.5 kg
Distance traveled, d = 25.7 m
Angle below the horizontal, θ = 30.9°
Frictional force, F_friction = 35.3 N
Step 1: Find the horizontal component of the force.
The horizontal component can be found using the formula: F_horizontal = F * cos(θ)
F_horizontal = F * cos(30.9°)
Step 2: Find the vertical component of the force.
The vertical component can be found using the formula: F_vertical = F * sin(θ)
F_vertical = F * sin(30.9°)
Since the cart is pushed at a constant velocity, the net force along the horizontal direction will be zero since the frictional force opposes the pushing force.
Step 3: Determine the work done by the pushing force.
The work done by a force can be calculated using the formula: Work = Force * Distance * cos(θ)
Work_pushing = F_horizontal * d * cos(0°)
Step 4: Determine the work done by the frictional force.
The work done by a force can be calculated using the formula: Work = Force * Distance * cos(θ)
Work_friction = F_friction * d * cos(180°)
Step 5: Determine the work done by the gravitational force.
The work done by the gravitational force can be calculated using the formula: Work = Force * Distance * cos(θ)
Work_gravitational = (m * g) * d * cos(90°)
Let's calculate the values.
Step 1: Find the horizontal component of the force.
F_horizontal = F * cos(30.9°)
F_horizontal = F * 0.866
Step 2: Find the vertical component of the force.
F_vertical = F * sin(30.9°)
F_vertical = F * 0.5
Since the cart is pushed at a constant velocity, the net force along the horizontal direction will be zero since the frictional force opposes the pushing force.
Step 3: Determine the work done by the pushing force.
Work_pushing = F_horizontal * d * cos(0°)
Work_pushing = F * 0.866 * 25.7 * cos(0°)
Work_pushing = F * 22.3349
Step 4: Determine the work done by the frictional force.
Work_friction = F_friction * d * cos(180°)
Work_friction = 35.3 * 25.7 * cos(180°)
Work_friction = -905.21
Step 5: Determine the work done by the gravitational force.
Work_gravitational = (m * g) * d * cos(90°)
Work_gravitational = (10.5 * 9.8) * 25.7 * cos(90°)
Work_gravitational = 2554.91
So, the magnitude of the force that the shopper exerts is F.
The work done by the pushing force is Work_pushing.
The work done by the frictional force is Work_friction.
The work done by the gravitational force is Work_gravitational.
To solve this problem, we need to consider the forces acting on the shopping cart and use the concepts of work and energy.
First, let's break down the forces acting on the cart:
1. Pushing Force (P): The force exerted by the shopper pushing the cart.
2. Frictional Force (Ff): The force opposing the motion of the cart.
3. Gravitational Force (Fg): The force due to the weight of the cart.
(a) To find the magnitude of the force that the shopper exerts (P), we need to consider the horizontal and vertical components of the force.
The horizontal component of the pushing force (Ph) can be found using the trigonometric relationship:
Ph = P * cos(θ)
where θ is the angle below the horizontal, given as 30.9°.
The vertical component of the pushing force (Pv) can be found using the same relationship:
Pv = P * sin(θ)
Since the cart is moving at a constant velocity, the net force acting on it is zero in the horizontal direction. Therefore, the magnitude of the pushing force is equal to the magnitude of the frictional force:
|Ph| = Ff
(b) The work done by the pushing force (Wp) is given by the equation:
Wp = Ph * d
where d is the distance traveled by the cart, given as 25.7 m.
(c) The work done by the frictional force (Wf) can be calculated using the equation:
Wf = -Ff * d
Negative sign indicates that the work done by the friction force is in the opposite direction of the displacement.
(d) The work done by the gravitational force (Wg) is equal to the change in potential energy. Since the cart is moving horizontally on a flat surface, there is no change in height, and therefore, no change in potential energy. Hence, Wg = 0.
Now, let's substitute the given values into the equations to find the answers:
(a) The magnitude of the pushing force:
|Ph| = Ff = 35.3 N
(b) The work done by the pushing force:
Wp = Ph * d = (P * cos(θ)) * d
(c) The work done by the frictional force:
Wf = -Ff * d = -35.3 N * 25.7 m
(d) The work done by the gravitational force:
Wg = 0
To find the values of Wp and Wf, we need the value of the pushing force (P). Unfortunately, the problem does not provide enough information to calculate the value of P. Therefore, we cannot determine the work done by the pushing force (Wp) and the work done by the frictional force (Wf).