To find the total normal force acting on the cart, we need to consider both the gravitational force and the vertical component of the applied force. Here’s a step-by-step solution:
1. Determine the mass of the cart:
The mass [tex]\( m \)[/tex] of the cart is given as [tex]\( 7.32 \)[/tex] kg.
2. Identify the force applied and its angle:
The force [tex]\( F \)[/tex] applied by the shopper is [tex]\( 14.7 \)[/tex] N, directed at [tex]\( -32.7^\circ \)[/tex] below the horizontal.
3. Calculate the gravitational force acting downward:
The gravitational force [tex]\( F_g \)[/tex] can be calculated using the relation:
[tex]\[
F_g = m \times g
\][/tex]
where [tex]\( g \)[/tex] is the acceleration due to gravity ([tex]\( 9.81 \ \text{m/s}^2 \)[/tex]):
[tex]\[
F_g = 7.32 \ \text{kg} \times 9.81 \ \text{m/s}^2 = 71.8092 \ \text{N}
\][/tex]
4. Determine the vertical component of the applied force:
Since the force is applied at an angle below the horizontal, we need the vertical component [tex]\( F_v \)[/tex]. The vertical component can be found using the sine of the angle [tex]\( \theta \)[/tex]:
[tex]\[
F_v = F \times \sin(\theta)
\][/tex]
where [tex]\( \theta = -32.7^\circ \)[/tex]. Converting this angle to radians:
[tex]\[
\theta_{\text{rad}} = -32.7^\circ \times \frac{\pi}{180} = -0.5707226654021458 \ \text{radians}
\][/tex]
Now, calculate the vertical component:
[tex]\[
F_v = 14.7 \ \text{N} \times \sin(-0.5707226654021458) = 14.7 \ \text{N} \times (-0.5707226654021458) = -7.941532711021528 \ \text{N}
\][/tex]
5. Calculate the total normal force:
The normal force [tex]\( n \)[/tex] is the sum of the gravitational force and the vertical component of the applied force:
[tex]\[
n = F_g + F_v
\][/tex]
Substituting the values:
[tex]\[
n = 71.8092 \ \text{N} + (-7.941532711021528 \ \text{N}) = 63.867667288978474 \ \text{N}
\][/tex]
Thus, the total normal force acting on the cart is:
[tex]\[
n = 63.867667288978474 \ \text{N}
\][/tex]
