If you stand on a bathroom scale, the spring inside the scale compresses 0.50 mm, and it tells you your weight is 680 N. Now if you jump on the scale from a height of 1.4 m, what does the scale read at its peak?
To determine what the bathroom scale reads at its peak when you jump on it from a height, we can follow these steps:
Step 1: Identify the key information given in the problem:
- The spring inside the scale compresses 0.50 mm.
- The scale reads 680 N when the spring is compressed.
Step 2: Recognize the principles involved:
- When you stand on the scale, your weight compresses the spring, and the scale measures the force exerted by the spring.
- When you jump on the scale from a height, both your weight and the force due to gravity are acting on the scale.
Step 3: Analyze the situation:
- When you jump on the scale, you will experience an initial force due to the compression of the spring from the height and an additional impact force due to your weight.
Step 4: Calculate the initial force due to the compression of the spring:
- The spring compression of 0.50 mm corresponds to the initial force acting on the scale.
- Using Hooke's Law, which states that the force exerted by a spring is proportional to the displacement, we can calculate the initial force:
F_initial = k * x
Where F_initial is the initial force, k is the spring constant, and x is the displacement.
- Since the scale reads 680 N when the spring is compressed, we can set up the equation:
680 N = k * 0.50 mm
Convert 0.50 mm to meters: 0.50 mm = 0.0005 m
Now we have the equation: 680 N = k * 0.0005 m
- Rearranging the equation, we find the spring constant:
k = 680 N / 0.0005 m
k = 1,360,000 N/m
Step 5: Calculate the maximum force on the scale from the height:
- When you jump from a height of 1.4 m, the potential energy is converted into kinetic energy before reaching the scale.
- The potential energy is given by the equation:
PE = m * g * h
Where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height.
- Rearranging the equation, we find the mass:
m = PE / (g * h)
Substituting the known values, we get:
m = 680 N / (9.8 m/s² * 1.4 m)
m ≈ 37.09 kg
Step 6: Calculate the total force on the scale at its peak:
- At the peak of your jump, both the initial force and the force due to your weight are acting on the scale.
- Since forces add up linearly, the total force on the scale at its peak is the sum of the initial force and the force due to your weight:
F_total = F_initial + (m * g)
F_total = (1,360,000 N/m * 0.0005 m) + (37.09 kg * 9.8 m/s²)
F_total = 680 N + 363.29 N
F_total ≈ 1043.29 N
Therefore, the scale would read approximately 1043.29 N at its peak when you jump on it from a height of 1.4 m.