This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Here are the solutions for 5.2.2 and 5.2.5: 5.2.2 The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy. 5.2.5 To calculate the magnitude of force F, we will use the Work-Energy Theorem, which states that the work done by non-conservative forces (W_nc) equals the change in the total mechanical energy ( E_mech). Since the incline is frictionless, the only non-conservative force doing work is F. Step 1: Identify the given values. Mass of the crate, m = 25 kg Initial speed at point Q, v_Q = 12 m/s Final speed at point R, v_R = 10.8 m/s Distance moved along the incline, d = 3.5 m Angle of inclination, = 30^ Acceleration due to gravity, g = 9.8 m/s^2 Step 2: Calculate the initial kinetic energy (KE_Q) at point Q. KE_Q = (1)/(2)mv_Q^2 KE_Q = (1)/(2)(25 kg)(12 m/s)^2 KE_Q = (1)/(2)(25)(144) J KE_Q = 1800 J Step 3: Calculate the final kinetic energy (KE_R) at point R. KE_R = (1)/(2)mv_R^2 KE_R = (1)/(2)(25 kg)(10.8 m/s)^2 KE_R = (1)/(2)(25)(116.64) J KE_R = 1458 J Step 4: Calculate the change in kinetic energy ( KE). KE = KE_R - KE_Q KE = 1458 J - 1800 J KE = -342 J Step 5: Calculate the change in gravitational potential energy ( PE_g). The vertical height gained, h = d (). h = (3.5 m)(30^) h = (3.5 m)(0.5) h = 1.75 m PE_g = mgh PE_g = (25 kg)(9.8 m/s^2)(1.75 m) PE_g = 428.75 J Step 6: Apply the Work-Energy Theorem for non-conservative forces. The work done by the applied force F (W_F) is the non-conservative work. W_F = KE + PE_g W_F = -342 J + 428.75 J W_F = 86.75 J Step 7: Calculate the magnitude of force F. The work done by force F is also given by W_F = Fd , where is the angle between the force and displacement. Since F is parallel to the incline and the displacement is up the incline, = 0^, so (0^) = 1. W_F = Fd F = (W_F)/(d) F = 86.75 J3.5 m F = 24.7857... N Rounding to three significant figures: F = 24.8 N The magnitude of force F is 24.8 N. Got more? Send 'em!