A person whose mass is m = 96.0 kg steps on a mechanical bathroom scale placed on an inclined plane that makes the angle α = 15.3° with the horizontal. What is the reading on the scale?
96 cos 15.3°
Think about it. Full weight at 0° (cos=1) and no weight at all at 90° (cos=0)
To find the reading on the scale, we need to determine the normal force acting on the person and convert it into a reading on the bathroom scale. Here are the steps to calculate it:
1. Draw a free-body diagram: It helps to visualize the forces acting on the person.
- Label the weight of the person as mg, where g is the acceleration due to gravity (approximately 9.8 m/s²).
- Draw a normal force (N) perpendicular to the inclined plane.
- Identify the angle α between the inclined plane and the horizontal.
2. Resolve the weight force into components: Since the inclined plane is at an angle to the horizontal, we need to resolve the weight force into two perpendicular components: one parallel to the inclined plane and one perpendicular to it.
- The component of the weight parallel to the inclined plane is mg*sin(α).
- The component of the weight perpendicular to the inclined plane is mg*cos(α).
3. Determine the normal force: The normal force (N) is the component of the weight perpendicular to the inclined plane.
- N = mg*cos(α)
4. Use the normal force to find the reading on the scale: The reading on the scale will be equal to the normal force (N) acting on the person.
- Reading on the scale = N = mg*cos(α)
Now, let's substitute the given values into the equation to find the reading on the scale:
m = 96.0 kg (The mass of the person)
α = 15.3° (The angle between the inclined plane and the horizontal)
Reading on the scale = N = mg*cos(α)
Reading on the scale = (96.0 kg) * (9.8 m/s²) * cos(15.3°)
Calculating the value, we get:
Reading on the scale = 935.16 N
Therefore, the reading on the bathroom scale would be 935.16 N.