A physics student pulls a block of mass m = 23 kg up an incline at a slow constant velocity for a distance of d = 4 m. The incline makes an angle q = 25° with the horizontal. The coefficient of kinetic friction between the block and the inclined plane is µk = 0.3.
a) What is the work Wm done by the student?
Wm = J
At the top of the incline, the string by which she was pulling the block breaks. The block, which was at rest, slides down a distance d = 4 m before it reaches a frictionless horizontal surface. A spring is mounted horizontally on the frictionless surface with one end attached to a wall. The block hits the spring, compresses it a distance L = 0.9 m, then rebounds back from the spring, retraces its path along the horizontal surface, and climbs up the incline. b) What is the speed v of the block when it first reaches the horizontal surface?
v = m/s
c) What is the spring constant k of the spring?
k = N/m
d) How far up the incline d1 does the block rebound?
To find the work done by the student in part (a), we first need to determine the force exerted by the student to maintain the constant velocity of the block up the incline.
1. Determine the gravitational force acting on the block:
The gravitational force, Fg, is given by the formula Fg = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2). Thus, Fg = 23 kg * 9.8 m/s^2 = 225.4 N.
2. Determine the normal force acting on the block:
The normal force, Fn, is the perpendicular component of the weight force acting on the block. Since the incline makes an angle of 25° with the horizontal, the normal force can be calculated as Fn = Fg * cos(q), where q is the angle of the incline. Thus, Fn = 225.4 N * cos(25°) = 203.1 N.
3. Determine the force of kinetic friction acting on the block:
The force of kinetic friction, Fk, can be calculated as Fk = µk * Fn, where µk is the coefficient of kinetic friction. Thus, Fk = 0.3 * 203.1 N = 60.93 N.
4. Determine the force exerted by the student:
Since the block is moving at a constant velocity, the force exerted by the student, Fs, must be equal in magnitude and opposite in direction to the force of kinetic friction. Thus, Fs = -60.93 N.
5. Calculate the work done by the student:
The work done by an applied force can be calculated as W = F * d * cos(θ), where F is the force applied, d is the displacement, and θ is the angle between the force and displacement vectors. In this case, since the force and displacement are in the same direction, cos(θ) = 1. Thus, Wm = Fs * d = -60.93 N * 4 m = -243.72 J.
Therefore, the work done by the student is Wm = -243.72 J.
Moving on to part (b), to find the speed of the block when it first reaches the horizontal surface:
1. Use the principle of conservation of mechanical energy:
At the top of the incline, the block has gravitational potential energy, which is converted to kinetic energy as it moves down. The work done by the spring will also be stored as potential energy. Therefore, we can use the principle of conservation of mechanical energy, expressed as:
m * g * d = (1/2) * m * v^2 + (1/2) * k * L^2
where m is the mass of the block, g is the acceleration due to gravity, d is the distance the block slides down, v is the velocity of the block when it reaches the horizontal surface, k is the spring constant, and L is the distance the spring is compressed.
2. Solve for v:
Rearranging the equation, we get:
v^2 = (2 * (m * g * d - k * L^2)) / m
Taking the square root, we have:
v = √((2 * (m * g * d - k * L^2)) / m)
Moving on to part (c), to find the spring constant k:
1. Use the information from part (b):
From part (b), we have the equation:
v = √((2 * (m * g * d - k * L^2)) / m)
Rearranging, we get:
k = ((2 * m * g * d) / L^2) - ((m * v^2) / L^2)
Finally, for part (d), to find the distance up the incline d1 to which the block rebounds:
1. Use the principle of conservation of mechanical energy:
The block starts with potential energy when it reaches the horizontal surface, and this is converted to kinetic energy as it moves up the incline. Therefore, we can again use the principle of conservation of mechanical energy as:
(1/2) * m * v^2 = m * g * d1 + (1/2) * k * L^2
where m is the mass of the block, v is the velocity of the block, g is the acceleration due to gravity, d1 is the distance up the incline, k is the spring constant, and L is the distance the spring is compressed.
2. Solve for d1:
Rearranging the equation, we find:
d1 = (v^2 - (k * L^2) / (2 * g))
Plug in the known values to find the specific answer to part (d).
Remember to use proper unit conversions and substitutions for values to get the final numerical answers.