Question

A ball of unknown mass m is tossed straight up with initial speed = 2.4 m/s. At the moment it is released, the ball is a height ℎ = 0.60 m above a spring-mounted platform, as shown in the figure. The ball rises, peaks, and falls back toward the platform, ultimately compressing the spring a maximum distance = 8.0 cm from its relaxed position. Assume that the spring constant is = 2500 N/m, and that the mass of the spring and platform is negligible. What is the mass of the ball, m, assuming that there is no friction or air resistance?

Answer 1

To find the mass of the ball, the conservation of energy principle is applied. The sum of the gravitational potential energy and kinetic energy at release equals the spring potential energy at maximum compression. The mass is calculated as 11.4 kg.

Step-by-step explanation:

To determine the mass of the ball, we can use the law of conservation of energy. The potential, kinetic, and elastic energy of the system must be equal at the point of release and at the maximum compression of the spring.

At the release point, the ball has gravitational potential energy (PEgrav) and kinetic energy (KE). At the maximum compression of the spring, the ball has spring potential energy (PEspring).

PEgrav + KE = PEspring

mgh + ½mv2 = ½kx2

Where:
m = mass of the ball
g = acceleration due to gravity (9.8 m/s2)
h = initial height (0.60 m)
v = initial speed (2.4 m/s)
k = spring constant (2500 N/m)
x = maximum compression of the spring (0.08 m)

Substituting the known values into the equation, we can solve for m:

m(9.8 m/s2 * 0.60 m) + ½m(2.4 m/s)2 = ½(2500 N/m)(0.08 m)2

5.88m + 2.88m = 100

8.76m = 100

m = 100 / 8.76

m = 11.4 kg

The mass of the ball is approximately 11.4 kg, considering the system is ideal and there's no friction or air resistance.