A block of mass m = 300 g is released from rest and slides down a frictionless track of height h = 59.4 cm. At the bottom of the track the block slides freely along a horizontal table until it hits a spring attached to a heavy, immovable wall. The spring compressed by 2.21 cm at the maximum compression. What is the value of the spring constant k?
PE=KE: mgh= KE
KE=PE(of compressed spring) KE= kx^2/2
mgh = kx^2/2
k =2mgh/x^2
To determine the value of the spring constant k, we can use the principle of conservation of mechanical energy.
First, let's find the gravitational potential energy the block has when it is at the top of the track. The gravitational potential energy is given by the formula:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (approximated as 9.8 m/s^2), and h is the height. Converting the mass to kilograms and the height to meters, we have:
PE = (0.3 kg) * (9.8 m/s^2) * (0.594 m)
Next, let's find the spring potential energy when the spring is compressed. The spring potential energy is given by the formula:
PE_spring = 0.5 * k * x^2
where k is the spring constant and x is the displacement (maximum compression). Converting the displacement to meters, we have:
PE_spring = 0.5 * k * (0.0221 m)^2
According to the conservation of mechanical energy, the initial potential energy should be equal to the final potential energy. Therefore:
PE = PE_spring
Substituting the values we have:
(0.3 kg) * (9.8 m/s^2) * (0.594 m) = 0.5 * k * (0.0221 m)^2
Simplifying the equation, we can calculate the value of k:
(0.3 kg) * (9.8 m/s^2) * (0.594 m) = 0.5 * k * (0.0221 m)^2
Solving for k:
k = 2 * (0.3 kg) * (9.8 m/s^2) * (0.594 m) / (0.0221 m)^2
k ≈ 500 N/m
Therefore, the value of the spring constant k is approximately 500 N/m.